A part of some device is a rotating one. Determine the longest time a motorcyclist will be in a cellular coverage area

Task 1. After rain, the water level in the well may rise. boy measuring time falling small pebbles into the well and calculates the distance to the water using the formula , where is the distance in meters, - fall time in seconds. Before the rain, the fall time of the pebbles was 1.2 s. How much must the water level rise after rain in order for the measured time to change by 0.2 s? Express your answer in meters.

Solution:

Calculate the distance to water before rain:

During rain, the water level will rise, the time of falling of the pebble will decrease and will be 1 s.

Then the distance to the water after the rain will be m.

Accordingly, the water level will rise after rain by m.

Answer: 2.2.

Task 2. The height above the ground of a ball tossed up changes according to the law , where is the height in meters, - time in seconds elapsed since the throw. How many seconds will the ball be at a height of at least 4 meters?

Solution:

We find the time of interest to us from the inequality:

The roots of the square trinomial: 0.2 and 2.4.

So we move on to the next inequality:

Therefore, the ball will be at a height of at least 4 meters for seconds.

Answer: 2.2.

Task 3. If you rotate a bucket of water on a rope in a vertical plane fast enough, then the water will not pour out. When the bucket rotates, the force of water pressure on the bottom does not remain constant: it is maximum at the bottom and minimum at the top. Water will not pour out if the force of its pressure on the bottom is positive at all points of the trajectory except the top, where it can be equal to zero. At the top point, the pressure force, expressed in newtons, is equal to , where is the mass of water in kilograms, is the speed of the bucket in m/s, is the length of the rope in meters, is the acceleration of free fall (count m/s). With what minimum speed must the bucket be rotated so that the water does not spill out if the length of the rope is 160 cm? Express your answer in m/s.

Solution:

Water will not pour out if the force of its pressure on the bottom is positive at all points of the trajectory except the top, where it can be equal to zero.

Don't forget to convert centimeters to meters!

Since is a positive value, we pass to an equivalent inequality:

Since the variable is non-negative, the inequality is equivalent to the following:

The smallest value corresponding to the inequality is 4.

Task 4. A crane is fixed in the side wall of a high cylindrical tank at the very bottom. After it is opened, water begins to flow out of the tank, while the height of the water column in it, expressed in meters, changes according to the law , where t is the time in seconds that has elapsed since the tap was opened, m is the initial height of the water column, is the ratio of the cross-sectional areas of the tap and the tank, and is the free fall acceleration (calculate m/s). In how many seconds after opening the faucet will a quarter of the original volume of water remain in the tank?

Solution:

The initial height of the column in the tank (at ) - m.

A quarter of the volume will remain in the tank when the height of the water column in the tank becomes m.

Substitute in the main formula:

Thus, 400 seconds after opening the tap, a quarter of the original volume of water will remain in the tank.

Answer: 400.

Task 5. The dependence of temperature (in degrees Kelvin) on time for a heating element of a certain device was obtained experimentally and, in the temperature range under study, is determined by the expression , where t- time in minutes, K, K/min, K/min. It is known that at a heater temperature above 1750 K, the device may deteriorate, so it must be turned off. Determine the maximum time after the start of work to turn off the device. Express your answer in minutes.

Solution:

Let's find corresponding to

Substituting all known values, we get:

In 2 minutes after turning on the device will heat up to 1750 K, and if it is heated further, the device may deteriorate.

Therefore, the device must be switched off after 2 minutes.

Task 6. To wind the cable at the factory, a winch is used, which winds the cable on a coil with uniform acceleration. The angle through which the coil turns changes with time according to the law , where - time in minutes, min - initial angular velocity rotation of the coil, and min - the angular acceleration with which the cable is wound. The worker must check the winding progress no later than the moment when the winding angle reaches 3000˚. Determine the time after the start of the winch, no later than which the worker must check its operation. Express your answer in minutes.

Solution:

Let's find , corresponding to the winding angle :

Minutes (due to the non-negativity of the variable we have one root

The worker must check the operation of the winch no later than 30 minutes after the start of work.

Task 7. A car moving at the initial moment of time with a speed of m/s starts braking with a constant acceleration of m/s. Per seconds after the start of braking, he traveled the distance (m). Determine the time elapsed from the start of braking, if it is known that during this time the car traveled 30 meters. Express your answer in seconds.

Solution:

Time according to the condition , elapsed from the start of braking, is found from the following equation:

In 2 seconds after braking, the car will cover a distance of 30 m.

Task 8. A part of some device is a rotating coil. It consists of three homogeneous coaxial cylinders: a central cylinder with mass kg and radius cm, and two lateral cylinders with mass kg and radii . In this case, the moment of inertia of the coil relative to the axis of rotation, expressed in kgcm, is given by the formula. At what maximum value of the moment of inertia of the coil does not exceed the limit value of 1300 kg cm? Express your answer in centimeters.

Solution:

The moment of inertia of the coil must not exceed the limit value of 1300 kg cm, therefore

Since , we get:

So, the maximum suitable value is 10 cm.

Task 9. At the shipyard, engineers are designing a new apparatus for diving to shallow depths. The design has the shape of a sphere, which means that the buoyant (Archimedean) force acting on the apparatus, expressed in newtons, will be determined by the formula: read N/kg). What can be the maximum radius of the apparatus so that the buoyancy force when immersed is no more than 42,000 N? Express your answer in meters.

Solution:

The buoyancy force when immersed should be no more than 30618 N, therefore

Accordingly, the maximum radius of the apparatus that meets the inequality is 1.

Task 10. To determine the effective temperature of stars, the Stefan–Boltzmann law is used, according to which the radiation power of a heated body P, measured in watts, is directly proportional to its surface area and the fourth power of temperature: , where is a constant, the area is measured in square meters, and the temperature is in degrees Kelvin. It is known that a certain star has an area m, and the power radiated by it is not less than watts. Determine the lowest possible temperature of this star. Give your answer in degrees Kelvin.

Solution:

Let's solve the inequality:

We reduce both sides of the inequality by

Multiply both sides by 128:

Since , we have:

The lowest possible temperature of a star is 4000 K.

Answer: 4000.

You can pass part 2.

1. The company sells its products at a price p=500 rub. per unit, the variable costs of producing one unit of output are rubles, the fixed costs of the enterprise f = 700,000 rubles. per month. The monthly operating profit of the enterprise (in rubles) is calculated by the formula . Determine the smallest monthly production q(units of production), in which the monthly operating profit of the enterprise will be at least 300,000 rubles. 5000

2. After rain, the water level in the well may rise. boy measuring time t falling small pebbles into the well and calculates the distance to the water using the formula h \u003d 5t 2, where h- distance in meters, t= fall time in seconds. Before the rain, the fall time of the pebbles was 0.6 s. How much must the water level rise after rain in order for the measured time to change by 0.2 s? Express your answer in meters. 1

3. The dependence of the volume of demand q(units per month) for the products of a monopoly enterprise from the price p(thousand rubles) is given by the formula q = 100 - 10p. Company revenue for the month r(in thousand rubles) is calculated by the formula . Determine the highest price p, at which the monthly revenue will be at least 240 thousand rubles. Give the answer in thousand rubles 6

4. The height above the ground of a ball tossed up changes according to the law , where h- height in meters t- time in seconds elapsed since the throw. How many seconds will the ball be at a height of at least three meters? 1,2

5. If you rotate a bucket of water on a rope in a vertical plane fast enough, then the water will not pour out. When the bucket rotates, the force of water pressure on the bottom does not remain constant: it is maximum at the bottom and minimum at the top. Water will not pour out if the force of its pressure on the bottom is positive at all points of the trajectory except the top, where it can be equal to zero. At the top point, the pressure force, expressed in newtons, is , where m is the mass of water in kilograms, v- the speed of the bucket in m / s, L- rope length in meters, g- free fall acceleration (calculate ). With what minimum speed must the bucket be rotated so that the water does not spill out if the length of the rope is 40 cm? Express your answer in m/s 2

6. A crane is fixed in the side wall of a high cylindrical tank at the very bottom. After it is opened, water begins to flow out of the tank, while the height of the water column in it, expressed in meters, changes according to the law , where t- time in seconds elapsed since the tap was opened, H 0 = 20 m - the initial height of the water column, - the ratio of the cross-sectional areas of the tap and the tank, and g- acceleration of gravity (). In how many seconds after opening the faucet will a quarter of the original volume of water remain in the tank? 5100

7. A crane is fixed in the side wall of a high cylindrical tank at the very bottom. After opening it, water begins to flow out of the tank, while the height of the water column in it, expressed in meters, changes according to the law, where m is the initial water level, m/min 2, and m/min are constants, t- time in minutes elapsed since the valve was opened. How long will water flow out of the tank? Give your answer in minutes 20

8. A stone-throwing machine shoots stones at some sharp angle to the horizon. The flight path of the stone is described by the formula , where m -1 , are constant parameters, x(m) - stone displacement horizontally, y(m) - the height of the stone above the ground. At what greatest distance (in meters) from a fortress wall 8 m high should a car be positioned so that the stones fly over the wall at a height of at least 1 meter? 90

9. The dependence of temperature (in degrees Kelvin) on time for a heating element of a certain device was obtained experimentally and, in the temperature range under study, is determined by the expression , where t- time in minutes, T 0 \u003d 1400 K, a \u003d -10 K / min 2, b \u003d 200 K / min. It is known that at a heater temperature above 1760 K, the device may deteriorate, so it must be turned off. Determine the maximum time after the start of work to turn off the device. Express your answer in minutes 2

10. To wind the cable at the factory, a winch is used, which winds the cable on a coil with uniform acceleration. The angle through which the coil turns changes with time according to the law , where t is the time in minutes, is the initial angular velocity of the coil, and is the angular acceleration with which the cable is wound. The worker must check the progress of its winding no later than the moment when the winding angle reaches 1200 0 . Determine the time after the start of the winch, no later than which the worker must check its operation. Express your answer in minutes. 20

11. A motorcyclist moving through the city at a speed of km/h leaves it and immediately after leaving it begins to accelerate with a constant acceleration a = 12 km/h. The distance from the motorcyclist to the city, measured in kilometers, is determined by the expression. Determine the longest time that a motorcyclist will be in a cellular service area if the operator guarantees coverage within a distance of no more than 30 km from the city. Express your answer in minutes 30

12. A car moving at the initial moment of time with a speed of m / s began braking with a constant acceleration a \u003d 5 m / s. Per t seconds after the start of braking, he traveled the distance (m). Determine the time elapsed from the start of braking, if it is known that during this time the car traveled 30 meters. Express your answer in seconds. 60

13. A part of some device is a rotating coil. It consists of three homogeneous coaxial cylinders: a central cylinder with mass m = 8 kg and radius R = 10 cm, and two side cylinders with mass M = 1 kg and radii R + h. In this case, the moment of inertia of the coil relative to the axis of rotation, expressed in kg. cm 2 is given by the formula . At what maximum value h the moment of inertia of the coil does not exceed the limit value of 625 kg. cm 2? Express your answer in centimeters. 5

14. At the shipyard, engineers are designing a new apparatus for diving to shallow depths. The design has a cubic shape, which means that the buoyancy force acting on the apparatus, expressed in newtons, will be determined by the formula: , where l is the length of the edge of the cube in meters, is the density of water, and g- free fall acceleration (assume g=9.8 N/kg). What can be the maximum length of the edge of the cube to ensure its operation in conditions where the buoyancy force when immersed will be no more than 78400N? Express your answer in meters. 2

15. At the shipyard, engineers are designing a new apparatus for diving to shallow depths. The design has the shape of a sphere, which means that the buoyant (Archimedean) force acting on the apparatus, expressed in newtons, will be determined by the formula: , where is a constant, r is the radius of the apparatus in meters, is the density of water, and g- free fall acceleration (assume g=10 N/kg). What can be the maximum radius of the apparatus so that the buoyancy force during immersion is no more than 336,000 N? Answer in meters 2

16. To determine the effective temperature of stars, the Stefan–Boltzmann law is used, according to which the radiation power of a heated body P, measured in watts, is directly proportional to its surface area and the fourth power of temperature: , where is a constant, area S measured in square meters, and the temperature T- in degrees Kelvin. It is known that a certain star has an area of ​​m 2, and the power radiated by it P not less than W. Determine the lowest possible temperature of this star. Give your answer in degrees Kelvin 4000

17. To obtain an enlarged image of a light bulb on the screen, a converging lens with a main focal length cm is used in the laboratory. the screen will be clear if the ratio is met. Indicate the smallest distance from the lens that a light bulb can be placed so that its image on the screen is clear. Express your answer in centimeters. 36

18. Before departure, the locomotive emitted a beep with a frequency of Hz. A little later, a locomotive approaching the platform blew a horn. Due to the Doppler effect, the frequency of the second beep f greater than the first: it depends on the speed of the locomotive according to the law (Hz), where c is the speed of sound in sound (in m/s). A person standing on the platform distinguishes signals by tone if they differ by at least 10 Hz. Determine the minimum speed with which the locomotive approached the platform if the person could distinguish the signals, and c = 315 m/s. Express your answer in m/s 7

19. According to Ohm's law for complete chain the current strength, measured in amperes, is equal to, where is the EMF of the source (in volts), Ohm is its internal resistance, R- circuit resistance (in ohms). At what minimum resistance of the circuit will the current strength be no more than 20% of the short circuit current strength? (Express your answer in ohms. 4

20. Current in the circuit I(in amperes) is determined by the voltage in the circuit and the resistance of the electrical appliance according to Ohm's law: , where U- voltage in volts, R- the resistance of the electrical appliance in ohms. A fuse is included in the mains, which melts if the current exceeds 4 A. Determine what the minimum resistance must be for an electrical appliance connected to a 220 volt outlet so that the network continues to work. Express your answer in ohms. 55

21. The amplitude of the pendulum oscillations depends on the frequency of the driving force, determined by the formula , where is the frequency of the driving force (in), is a constant parameter, is the resonant frequency. Find the maximum frequency , less than the resonant one, for which the oscillation amplitude exceeds the value by no more than 12.5%. Express your answer in 120

22. Devices are connected to the power outlet, the total resistance of which is ohms. In parallel with them, an electric heater is supposed to be connected to the outlet. Determine the smallest possible resistance of this electric heater if it is known that when two conductors with resistances Ohm and Ohm are connected in parallel, their total resistance is given by the formula (Ohm), and for the normal functioning of the electrical network, the total resistance in it must be at least 9 Ohm. Express your answer in ohms. 10

23. The coefficient of performance (COP) of some engine is determined by the formula , where is the temperature of the heater (in degrees Kelvin), is the temperature of the refrigerator (in degrees Kelvin). At what minimum temperature of the heater will the efficiency of this engine be at least 15% if the refrigerator temperature is K? Express your answer in degrees Kelvin. 400

24. The coefficient of efficiency (COP) of a feed steamer is equal to the ratio of the amount of heat spent on heating water with a mass (in kilograms) from temperature to temperature (in degrees Celsius) to the amount of heat obtained from burning firewood with a mass of kg. It is determined by the formula, where J / (kg K) is the heat capacity of water, J / kg is the specific heat of combustion of firewood. Determine the smallest amount of firewood that will need to be burned in the feed steamer to heat a kg of water from 10 0 C to boiling, if it is known that the efficiency of the feed steamer is not more than 21%. Answer in kilograms 18

25. The support shoes of a walking excavator with a mass of tons are two hollow beams meters long and wide. s meters each. The pressure of the excavator on the soil, expressed in kilopascals, is determined by the formula, where m- weight of the excavator (in tons), l- the length of the beams in meters, s- beam width in meters, g- free fall acceleration (read m/s). Determine the smallest possible width of the support beams if it is known that the pressure p should not exceed 140 kPa. Express your answer in meters. 2,5

26. To a source with EMF V and internal resistance Ohm, they want to connect a load with resistance R Ohm. The voltage across this load, expressed in volts, is given by . At what the smallest value load resistance voltage on it will be at least 50 V? Express your answer in ohms. 5

27. When approaching the source and receiver of sound signals moving in a certain medium in a straight line towards each other, the frequency of the sound signal recorded by the receiver does not coincide with the frequency of the original signal Hz and is determined by the following expression: (Hz), where c is the speed of signal propagation in the medium (in m/s), and m/s and m/s are the velocities of the receiver and source relative to the medium, respectively. At what maximum speed c(in m/s) signal propagation in the medium signal frequency at the receiver f will be at least 160 Hz 390

28. The locator of a bathyscaphe, evenly plunging vertically downward, emits ultrasonic pulses with a frequency of 749 MHz. The speed of descent of the bathyscaphe, expressed in m/s, is determined by the formula, where m/s is the speed of sound in water, is the frequency of the emitted pulses (in MHz), f- frequency of the signal reflected from the bottom, recorded by the receiver (in MHz). Determine the highest possible frequency of the reflected signal f if the bathyscaphe sinking speed should not exceed 2 m/s 751

29. l km with constant acceleration , is calculated by the formula . Determine the minimum acceleration with which the car must move in order to travel one kilometer and acquire a speed of at least 100 km / h. Express your answer in km/h 5000

30. When a rocket moves, its visible length for a stationary observer, measured in meters, is reduced according to the law , where m is the length of the resting rocket, km/s is the speed of light, and v- rocket speed (in km/s). What should be the minimum speed of the rocket so that its observed length becomes no more than 4 m? Express your answer in km/s 180000

31. The speed of a car accelerating from the starting point along a straight line segment of length l km with constant acceleration a km/h is calculated by the formula . Determine with what minimum speed the car will move at a distance of 1 kilometer from the start, if, according to the design features of the car, the acceleration acquired by it is not less than 5000 km / h. Express your answer in km/h 100

32. It is planned to use a cylindrical column to support the canopy. Pressure P(in pascals), provided by a canopy and a column on a support, is determined by the formula, where m \u003d 1200 kg is the total mass of the canopy and column, D- diameter of the column (in meters). Assuming the free fall acceleration g=10 m/s, a, determine the smallest possible diameter of the column if the pressure exerted on the support should not exceed 400,000 Pa. Express your answer in meters. 0,2

33. A car whose mass is equal to m = 2160 kg starts moving with an acceleration that during t seconds remains unchanged, and during this time the path S = 500 meters passes. The value of the force (in newtons) applied to the car at this time is . Determine the longest time after the start of the movement of the car, for which it will cover the specified path, if it is known that the force F applied to the car, not less than 2400 N. Answer in seconds 30

34. In an adiabatic process, for an ideal gas, the law is satisfied, where p- gas pressure in pascals, V- volume of gas in cubic meters. In the course of an experiment with a monatomic ideal gas (for it ) from the initial state, in which Pa , the gas begins to compress. What is the largest volume V can occupy gas at pressures p not lower than Pa? Express your answer in cubic meters. 0,125

35. During the decay of a radioactive isotope, its mass decreases according to the law , where is the initial mass of the isotope, t(min) - elapsed time from the initial moment, T- half-life in minutes. In the laboratory, a substance was obtained containing at the initial moment of time mg of the isotope Z, whose half-life is min. In how many minutes will the mass of the isotope be at least 5 mg 30

36. The process equation in which the gas participated is written as , where p(Pa) - gas pressure, V- volume of gas in cubic meters, a is a positive constant. For what is the smallest value of the constant a halving the volume of gas involved in this process leads to an increase in pressure by at least 4 times 2

37. The installation for demonstrating adiabatic compression is a vessel with a piston that sharply compresses the gas. In this case, the volume and pressure are related by the relation , where p(atm.) - pressure in the gas, V- volume of gas in liters. Initially, the volume of the gas is 1.6 liters, and its pressure is one atmosphere. In accordance with the technical specifications, the pump piston can withstand a pressure of not more than 128 atmospheres. Determine the minimum volume the gas can be compressed to. Express your answer in liters. 0,05

38. The capacitance of the high-voltage capacitor in the TV F. A resistor with an ohm resistance is connected in parallel with the capacitor. During operation of the TV, the voltage on the capacitor is kV. After turning off the TV, the voltage across the capacitor decreases to a value U(kV) for the time defined by the expression (s), where is a constant. Determine (in kilovolts) the highest possible voltage across the capacitor if at least 21 seconds have passed since the TV was turned off 2

39. To heat a room, the temperature in which is equal to, hot water is passed through a heating radiator with a temperature of . Consumption of water passing through the pipe kg / s. Passing through the pipe distance x(m), water is cooled to a temperature, and (m), where is the heat capacity of water, is the heat transfer coefficient, and is a constant. To what temperature (in degrees Celsius) will the water cool if the length of the pipe is 84 m 30

40. A diving bell, containing at the initial moment of time a mole of air with a volume of l, is slowly lowered to the bottom of the reservoir. In this case, isothermal compression of air to a final volume occurs. The work done by water when air is compressed is determined by the expression (J), where is constant, and K is the air temperature. What volume (in liters) will air take up if 10350 J of work was done during gas compression 8

41. A diving bell in the water, containing moles of air at atmospheric pressure, is slowly lowered to the bottom of the reservoir. In this case, isothermal compression of air occurs. The work done by water when air is compressed is determined by the expression (J), where is a constant, K is the air temperature, (atm) is the initial pressure, and (atm) is the final air pressure in the bell. To what maximum pressure can the air in the bell be compressed if the work done by compressing the air is not more than 6900 J? Give your answer in atmospheres 6

42. The ball is thrown at an angle to a flat horizontal surface of the ground. The flight time of the ball (in seconds) is determined by the formula . What is the smallest value of the angle (in degrees) for which the flight time will be at least 3 seconds if the ball is thrown with an initial speed of m/s? Assume that the free fall acceleration m/s 30

43. A part of some device is a square frame with a wire wound around it, through which a direct current is passed. The frame is placed in a uniform magnetic field so that it can rotate. The moment of the Ampere force tending to rotate the frame, (in N m) is determined by the formula, where is the current strength in the frame, Tl is the value of induction magnetic field, m is the size of the frame, is the number of turns of wire in the frame, a is the acute angle between the perpendicular to the frame and the induction vector. At what is the smallest value of the angle a (in degrees) the frame can start to rotate, if this requires that the unwinding moment M was not less than 0.75 N m 30

44. The sensor is designed in such a way that its antenna catches a radio signal, which is then converted into an electrical signal that changes over time according to the law , where is time in seconds, amplitude B, frequency , phase . The sensor is configured so that if the voltage in it is not lower than V, the lamp lights up. What part of the time (in percent) during the first second after the start of work will the light bulb be on 50

45. A very light charged metal ball with a charge of C rolls down a smooth inclined plane. At the moment when its speed is m / s, a constant magnetic field begins to act on it, the induction vector B which lies in the same plane and makes an angle a with the direction of motion of the ball. The value of the field induction Tl. In this case, the Lorentz force acts on the ball, equal to (N) and directed upwards perpendicular to the plane. What is the smallest value of the angle at which the ball will break away from the surface, if this requires that the force be not less than N? Give your answer in degrees 30

46. A small ball is thrown at an acute angle to a flat horizontal surface of the earth. The maximum flight height of the ball, expressed in meters, is determined by the formula, where m / s is the initial speed of the ball, and g- free fall acceleration (calculate m/s 2). What is the smallest value of the angle (in degrees) for the ball to fly over a wall 4 m high at a distance of 1 m 30

47. A small ball is thrown at an acute angle a to a flat horizontal surface of the earth. The distance that the ball flies is calculated by the formula (m), where m / s is the initial speed of the ball, and g- free fall acceleration (m/s 2). What is the smallest angle (in degrees) at which the ball will fly over a river 20 m wide 15

48. A flat closed circuit with an area of ​​S=0.5 m 2 is in a magnetic field, the induction of which increases uniformly. However, according to the law electromagnetic induction Faraday in the circuit appears EMF induction, the value of which, expressed in volts, is determined by the formula, where a is the acute angle between the direction of the magnetic field and the perpendicular to the contour, T / s is a constant, S- the area of ​​the closed circuit, located in the magnetic field (in m). At what minimum angle a (in degrees) will the induction emf not exceed V 60

49. The tractor pulls the sled with a force F = 80 kN directed at an acute angle a to the horizon. The work of the tractor (in kilojoules) on a section of length S = 50m is calculated by the formula . At what maximum angle a (in degrees) will the work done be at least 2000 kJ 60

50. The tractor pulls the sled with a force F=50 kN directed at an acute angle a to the horizon. Tractor power (in kilowatts) at speed v= 3 m/s is equal to . At what maximum angle a (in degrees) will this power be at least 75 kW 60

51. Under normal incidence of light with a wavelength of nm on a diffraction grating with a period d nm, a series of diffraction maxima are observed. In this case, the angle (measured from the perpendicular to the grating) at which the maximum is observed and the number of the maximum k related by the ratio. At what minimum angle (in degrees) can one observe the second maximum on a grating with a period not exceeding 1600 nm 30

52. Two bodies of mass kg each move with the same speed m/s at an angle to each other. The energy (in joules) released during their absolutely inelastic collision is determined by the expression . At what smallest angle (in degrees) must the bodies move so that at least 50 joules are released as a result of the collision. 60

53. The boat must cross a river with a width of m and with a current speed of u = 0.5 m/s so as to land exactly opposite the place of departure. It can move at different speeds, while the travel time, measured in seconds, is determined by the expression , where a is an acute angle that specifies the direction of its movement (counted from the coast). At what minimum angle a (in degrees) must one swim so that the travel time is no more than 200 s 45

54. A skateboarder jumps onto a platform standing on rails with a speed v = 3 m/s at an acute angle to the rails. From the push, the platform starts moving at a speed (m/s), where m = 80 kg is the mass of the skateboarder with the skateboard, and M = 400 kg is the mass of the platform. What is the maximum angle (in degrees) you need to jump at to accelerate the platform to at least 0.25 m/s? 60

55. A load with a mass of 0.08 kg oscillates on a spring with a speed that varies according to the law , where t- time in seconds. The kinetic energy of the load, measured in joules, is calculated by the formula , where m- mass of cargo (in kg), v- load speed (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 5 . 10 -3 J. Express your answer decimal round to hundredths if necessary. 0,25

56. A weight of 0.08 kg oscillates on a spring with a speed that varies according to the law , where t- time in seconds. The kinetic energy of the load is calculated by the formula , where m- mass of cargo (in kg), v- load speed (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 5 . 10 -3 J. Express your answer as a decimal fraction, if necessary, round to hundredths 0,25

57. The speed of the load oscillating on the spring changes according to the law (cm / s), where t- time in seconds. What fraction of the time from the first second did the speed exceed 2.5 cm/s? Express your answer as a decimal, round to hundredths if necessary. 0,17

58. The distance from an observer located at a low altitude of kilometers above the earth to the horizon line he observes is calculated by the formula , where (km) is the radius of the Earth. From what height is the horizon visible at a distance of 4 kilometers? Express your answer in kilometers.

59. An independent agency intends to introduce a rating of news publications based on indicators of informativeness, efficiency and objectivity of publications. Each indicator is evaluated with integers from -2 to 2.

The analyst making up the formula believes that the information content of publications is valued three times, and objectivity is twice as expensive as efficiency. As a result, the formula will take the form

What should be the number for the publication with the highest scores to be rated 30?

where is the average rating of the store by customers (from 0 to 1), is the rating of the store by experts (from 0 to 0.7) and is the number of buyers who rated the store.

61. An independent agency intends to introduce a rating of online news publications based on assessments of informativeness, efficiency, objectivity of publications, as well as the quality of the site. Each individual indicator is evaluated by readers on a 5-point scale with integers from 1 to 5.

What should be the number , so that the publication, which has all the highest ratings, would receive a rating of 1?

62. An independent agency intends to introduce a rating of online news publications based on assessments of informativeness, efficiency, objectivity of publications, as well as the quality of the site. Each individual indicator is evaluated by readers on a 5-point scale with integers from -2 to 2.

If for all four indicators a certain publication received the same rating, then the rating should coincide with this rating. Find the number at which this condition will be met.

Answer.8.

№ 5.2.(523). The height above the ground of a ball tossed up changes according to the law h(t) =1,6 + 8t – 5t 2 , where h- height in meters, t- time in seconds elapsed since the throw. How many seconds will the ball be at a height of at least 3 meters?

Solution. According to the condition of the problem, the ball will be at a height of at least 3 m, which means that the inequality h ≥ 3 or 1.6 + 8 t – 5t 2 ≥ 3.

Let's solve the resulting inequality: - 5 t 2 +8t – 1,4 ≥ 0; 5t 2 - 8t +1,4 ≤ 0.

Solve Equation 5 t 2 - 8t +1,4 = 0.

D= b 2 - 4ac= 8 2 - 4∙5∙1,4 = 64 - 28 = 36.

t 1,2 = = .

t 1 = = 0,2 , t 2 = 1,4.

5(t-0,2)(t- 1,4) ≤ 0; 0,2 ≤ t ≤ 1,4.

The ball was at a height of at least 3 m from the time 0.2 s to the time 1.4 s, that is, in the time period 1.4 - 0.2 = 1.2 (s).

Answer.1,2.

№ 5.3(526). If you rotate a bucket of water on a rope in a vertical plane fast enough, then the water will not pour out. When the bucket rotates, the force of water pressure on the bottom does not remain constant: it is maximum at the bottom point and minimum at the top. Water will not pour out if the force of its water pressure on the bottom is positive at all points of the trajectory, except for the top, where it can be equal to zero. At the top point, the pressure force, expressed in pascals, is equal to P \u003d m, where m is the mass of water in kilograms, is the speed of the bucket in m / s, L is the length of the rope in meters, g is the acceleration of free fall (take g = 10m / c 2). With what minimum speed should the bucket be rotated so that the water does not spill out if the length of the rope is 90 cm? Express your answer in m/s.

Solution. By the condition of the problem, P ≥ 0 or m ≥ 0.

Taking into account the numerical values ​​L= 90 cm = 0.9 m, g = 10m/s 2 and m 0, the inequality takes the form: - 10 ≥ 0; 2 ≥ 9.

Based physical sense problem ≥ 0, so the inequality takes the form

≥ 3. The smallest solution to the inequality = 3(m/s).

№ 5.4 (492). The dependence of temperature (in degrees Kelvin) on time (in minutes) for the heating element of a certain device was obtained experimentally and is given by the expression T( t) = T0 + bt + at 2, where T 0 = 1350 K, a\u003d -15 K / min 2, b = 180 K / min. It is known that at a heater temperature above 1650 K the device may deteriorate, so it must be turned off. Determine (in minutes) how long after the start of work you need to turn off the device?

Solution. Obviously, the device will operate at T( t) ≤ 1650 (K), that is, the inequality must be satisfied: T 0 + bt + at 2 ≤ 1650. Taking into account the numerical data T 0 = 1350K, a\u003d -15K / min 2, b = 180K/min, we have: 1350 + 180 t - 15 t 2 ≤ 1650; t 2 - 12t + 20 ≥ 0.

The roots of a quadratic equation t 2 - 12t + 20 = 0: t 1 =2 , t 2 =10.

Solution of the inequality: t ≤ 2, t ≥10.

According to the meaning of the problem, the solution of the inequality takes the form: 0 ≤ t ≤ 2, t ≥10.

The heater must be turned off after 2 minutes.

Answer. 2.

№ 5.5 (534). A stone-throwing machine shoots stones at some sharp angle to the horizon. The flight path of the stone is described by the formula y = ax 2 + bx, where a = - m -1, b = - constant coefficients, x(m) is the horizontal displacement of the stone, y(m) is the height of the stone above the ground. At what maximum distance (in meters) from a 9 m high fortress wall should a car be positioned so that the stones fly over the wall at a height of at least 1 meter?

Solution. According to the condition of the problem, the height of the stone above the ground will be at least 10 meters (the height of the wall is 9 m and above the wall is at least 1 meter), therefore, the inequality y ≥ 10 or ax 2 + bx ≥ 10. Including numerical data a = - m -1, b = inequality will take the form: - x 2 + x ≥ 10; x 2 - 160x + 6000 ≤ 0.

The roots of a quadratic equation x 2 - 160x + 6000 = 0 are the values x 1 = 60 and x 2 = 100.

(x - 60)(x - 100) ≤ 0; 60 ≤ x ≤ 100.

The largest solution to the inequality x= 100. The stone-throwing machine must be placed at a distance of 100 meters from the fortress wall.

Answer.100.

№ 5.6 (496). To wind the cable at the factory, a winch is used, which winds the cable on a reel with uniform acceleration. The angle at which the coil turns is measured over time according to the law = + , where = 20/min is the initial angular velocity of the coil, and = 8/min 2 is the angular acceleration with which the cable is wound. The worker must check the winding progress no later than the winding angle reaches 1200. Determine the time (in minutes) after the start of the winch, no later than which the worker must check its work.

Solution. The worker may not check the cable winding progress until the moment when the winding angle ≤ 1200, i.e. + ≤ 1200. Taking into account the fact that = 20/min, = 8/min 2, the inequality will take the form: + ≤ 1200.

20t + 4t 2 ≤ 1200; t2 + 5t – 300 ≤ 0.

Let's find the roots of the equation t 2 + 5t - 300 = 0.

According to the theorem, the inverse of Vieta's theorem, we have: t 1 ∙ t 2 = - 300, t 1 + t 2 = -5.

From: t 1 \u003d -20, t 2 \u003d 15.

Let's return to the inequality: (t +20)(t - 15) ≤ 0, whence -20 ≤ t ≤ 15, taking into account the meaning of the problem (t ≥ 0), we have: 0 ≤ t ≤ 15.

The worker must check the operation of the winch no later than 15 minutes after the start of its operation.

Answer. 15.

№ 5.7 (498). A motorcyclist moving through the city at a speed of 0 = 58 km/h leaves it and immediately after the exit begins to accelerate with constant acceleration a\u003d 8 km / h 2. The distance from the motorcyclist to the city is given by S= 0 t+ . Determine the longest time (in minutes) that a motorcyclist will be in a cellular service area if the operator guarantees coverage within a distance of 30 km from the city.

Solution. The motorcyclist will remain within the cellular coverage area for as long as S≤ 30, i.e. 0 t + ≤ 30. Considering that = 58 km / h, a= 8 km/h 2 inequality will take the form: 58 t + ≤ 30 or 58 t + 4t 2 - 30 ≤ 0.

Let's find the roots of the equation 4t 2 + 58t - 30 = 0.

D \u003d 58 2 - 4 4 ∙ (-30) \u003d 3364 + 480 \u003d 3844.

t 1 \u003d \u003d 0.5; t 2 = = - 15.

Let's return to the inequality: (t - 0.5)(t + 15) ≤ 0, whence -15 ≤ t ≤ 0.5, taking into account the meaning of the problem (t ≥ 0), we have: 0 ≤ t ≤ 0.5.

The motorcyclist will be in the zone of cellular communication for 0.5 hour or 30 minutes.

Answer.30.

№ 5.8 (504). A part of some device is a rotating coil. It consists of three homogeneous coaxial cylinders: a central one with a mass m = 4 kg and a radius R = 5 cm, two lateral cylinders with a mass M = 2 kg and a radius R + h each. In this case, the moment of inertia of the coil (in kg ∙ cm 2) relative to the axis of rotation is determined by the expression I \u003d + M (2Rh + h 2). At what maximum value (in cm) does the moment of inertia of the coil not exceed its limit of 250 kg ∙ cm 2?

Solution. According to the condition of the problem, the moment of inertia of the coil relative to the axis of rotation does not exceed the limit value of 250 kg ∙ cm 2, therefore, the inequality is fulfilled: I ≤ 250, i.e. + M (2Rh + h 2) ≤ 250. Taking into account the fact that m = 4 kg, R = 5 cm, M = 2 kg, the inequality will take the form: + 2∙ (2∙5∙h + h 2) ≤ 250 After simplification, we have:

h 2 +10h – 150 ≤ 0.

Let's find the roots of the equation h 2 +10 h - 75 = 0.

According to the theorem, the inverse of Vieta's theorem, we have: h 1 ∙ h 2 = - 75, h 1 + h 2 = -10.

From: t 1 \u003d -15, t 2 \u003d 5.

Let's return to the inequality: (t +15)(t - 5) ≤ 0, whence -15 ≤ t ≤ 5, taking into account the meaning of the problem (t ≥ 0), we have: 0 ≤ t ≤ 5.

The moment of inertia of the coil relative to the axis of rotation does not exceed the limit value of 250 kg ∙ cm 2 with a maximum h = 5 cm.

Answer. 5.

№ 5.9(502). A car moving at the initial moment of time with a speed of 0 = 21 m/s and decelerating with constant acceleration a\u003d 3 m / s 2, for the time t seconds after the start of braking, the path passes S= 0 t - . Determine (in seconds) the shortest time elapsed since the start of braking, if it is known that during this time the car has traveled at least 60 meters.

Solution. Since the car has traveled at least 60 meters after the start of braking, then S≥ 60, that is 0 t - ≥ 60. Considering that = 21 m / s, a= 3 m/s 2 the inequality will take the form:

21t - ≥ 60 or 42 t - 3t 2 - 120 ≥ 0, 3t 2 - 42t + 120 ≤ 0, t 2 - 14t + 40 ≤ 0.

Let's find the roots of the equation t 2 - 14t + 40 = 0.

According to the theorem converse to the Vieta theorem, we have: t 1 ∙ t 2 = 40, t 1 + t 2 = 14.

From: t 1 = 4, t 2 = 10.

Let's return to the inequality: (t - 4)(t - 10) ≤ 0, whence 4 ≤ t ≤ 10.

The shortest time elapsed from the start of braking is t = 4s.

Answer.4.

Literature.

USE: 3000 tasks with answers in mathematics. All tasks of group B / A.L. Semenov, I. V. Yashchenko and others / ed. A.L. Semenova, I.V. Yashchenko - M.; Publishing house "Exam". 2013

The optimal bank of tasks for preparing students. USE 2014. Mathematics. Tutorial. / A.V. Semenov, A. S. Trepalkin, I. V. Yashchenko and others / ed. I. V. Yashchenko; Moscow Center for Continuous Mathematical Education. - M.; Intellect Center, 2014

Koryanov A.G., Nadezhkina N.V. . Tasks B12. Application Content Tasks

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