7 Calculus Questions

1.

Newton's Law of Gravitation says that the magnitude Fof the force exerted by a body of mass m on a body of mass M is _ GmM ,2 where G is the gravitational constant and r is the distance between the bodies. (a) Find dF/dr. i: _2Gm_M dr 7'3 F J What is the meaning of dF/dr? dF/dr represents the rate of change of the distance between the bodies with respect to the force. ' dF/dr represents the rate of change of the mass with respect to the force. ' dF/dr represents the rate of change of the mass with respect to the distance between the bodies. dF/dr represents the amount of force per distance. 0 dedr represents the rate of change of the force with respect to the distance between the bodies. W What does the minus sign indicate? The minus sign indicates that as the distance between the bodies decreases, the magnitude of the force remains constant. The minus sign indicates that the force between the bodies is decreasing. o The minus sign indicates that as the distance between the bodies increases, the magnitude of the force decreases. ' The minus sign indicates that as the distance between the bodies increases, the magnitude of the force increases. ' The minus sign indicates that the bodies are being forced in the negative direction. (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 4 N/km when r = 20,000 km. How fast does this force change when r = 10,000 km? 32 x N/km At noon, ship A is 120 km west of ship B. Ship A is sailing east at 25 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 4:00 PM? A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.4 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? Part 1 of 4 Using the diagram below, find the relation between x and y. X IN = X y 2 x 12 - x 12A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.3 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall? (That is, nd the angle's rate of change when the bottom of the ladder is 6 ft from the wall.) x rad/s wall 10 A plane flying horizontally at an altitude of 2 mi and a speed of 570 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 3 mi away from the station. (Round your answer to the nearest whole number.) _________________________________________________________________________________________________________________________________________ Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 800 cm3, the pressure is 100 kPar and the pressure is increasing at a rate of 30 kPa/min. At what rate is the volume decreasing at this instant? x cm3/min Tutorial Exercise A cylindrical tank with radius 4 m is being filled with water at a rate of 3 m3/min. How fast is the height of the water increasing? Part 1 of 3 If h is the water's height, the volume of the water is V = x/2h. We must find dV/dt. Differentiating both sides of the equation gives the following. dv : 1/2 dt X Submit Skip (you cannot come back)