L3 Differentiation
Question Bank
A spherical hailstone grows as it falls through a storm cloud. Its volume is increasing at a rate of 3 mm³ per second.
Find the rate at which the radius of the hailstone is increasing at the instant its diameter is 12 mm.
The volume of a sphere is $V = \frac{4}{3}\pi r^3.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A bucket is being filled from a tap at a rate of 100 cm³ per second. When the water in the bucket is $h
nbsp;cm deep, the volume of water, in cm³, is$ V = \frac{5\pi}{3}\left[(10 + 0.2h)^3 - 1000\right]. $
Find the rate at which the water level is rising when the water is 15 cm deep. Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
At a quarry, a conveyor belt drops gravel onto a pile at a rate of 1.2 m³ per minute. The pile is always a cone whose base radius is 1.5 times its height.
Find the rate at which the height of the pile is increasing when the pile is 4 m high.
The volume of a cone is $V = \frac{1}{3}\pi r^2 h.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
Zoe blows through a straw into a puddle of soapy water on a flat table, making a bubble in the shape of a hemisphere. Air goes into the bubble at a constant rate of 15 cm³ per second.
Find the rate at which the radius of the bubble is increasing at the instant the volume of the bubble is $144\pi\text{ cm}^3$.
The volume of a hemisphere of radius $r$ is $V = \frac{2}{3}\pi r^3.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A street light is at the top of a pole 6 m tall. Sione, who is 1.8 m tall, walks in a straight line directly away from the pole at 1.4 m/s.
(a) Find the rate at which the length of Sione's shadow is increasing.
(b) Find the speed at which the tip of his shadow is moving along the ground.
You must use calculus and show any derivatives that you need to find when solving this problem.
A drone takes off from level ground and rises vertically at a steady 2.5 m/s. Arjun is standing 120 m from the take-off point, watching it.
Find the rate at which the angle of elevation of the drone from Arjun is increasing at the instant the drone is 90 m above the ground. Ignore Arjun's height.
Give your answer in radians per second, correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A funnel is an inverted cone, 12 cm across the top and 15 cm deep. It is full of oil, which drains out through the spout at a rate of 3 cm³ per second.
Find the rate at which the oil level is falling when the oil is 10 cm deep. Ignore the spout.
The volume of a cone is $V = \frac{1}{3}\pi r^2 h.$ Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
A crane on a wharf has a straight jib 30 m long, hinged at its lower end. The jib is being raised so that the angle $\theta$ it makes with the horizontal increases at 0.02 radians per second.
Find the rate at which the tip of the jib is rising at the instant the tip is 18 m above the level of the hinge.
You must use calculus and show any derivatives that you need to find when solving this problem.
A bath bomb is a sphere 7 cm in diameter when it is dropped into a bath. As it fizzes away it stays a sphere, and its radius decreases at a constant 0.02 cm per second.
Find the rate at which the volume of the bath bomb is decreasing at the instant it has lost half of its original volume.
The volume of a sphere is $V = \frac{4}{3}\pi r^3.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A boat is being pulled towards a wharf by a rope tied to its bow. The rope runs up to a winch on the wharf, 2.5 m higher than the bow, which winds the rope in at 0.5 m/s.
Find the speed at which the boat is approaching the wharf at the instant there is 6.5 m of rope between the bow and the winch. Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A stock trough is being filled with water at a rate of 6 litres per minute. When the water is $h
nbsp;cm deep ($0 \le h \le 30$), the volume of water in the trough is$ V = 150h(40 + h)\text{ cm}^3. $
Find the exact rate at which the water level is rising at the instant the trough holds 120 litres of water. (1 litre = 1000 cm³.)
You must use calculus and show any derivatives that you need to find when solving this problem.
A farm header tank is an inverted cone, 2 m across the top and 1.5 m deep. The tank is empty when water starts being pumped into it at a constant rate of 0.04 m³ per minute.
Find the rate at which the depth of the water is increasing 15 minutes after the pumping starts.
The volume of a cone is $V = \frac{1}{3}\pi r^2 h.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A lighthouse stands on a small island 1.5 km from a straight coastline. P is the point on the coast closest to the lighthouse. The lighthouse's beam turns at a constant 4 revolutions per minute, and as it sweeps past it lights up a spot that moves along the coast.
(a) Find the speed of the spot of light along the coast at the instant it is 2 km from P. Give your answer correct to 3 significant figures.
(b) Find the length of the stretch of coast, on both sides of P, where the spot of light moves at less than three times its speed at P. Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
At a pulp mill, wood chips are blown out of a pipe onto a heap at a rate of 3 m³ per minute. The heap is always a cone, and its sloping sides always make an angle of 45° with the ground.
The heap has base radius $r$, height $h$ and slant height $l$, all in metres.
Find the rate at which the slant height of the heap is increasing at the instant the radius of the heap is 6 m.
The volume of a cone is $V = \frac{1}{3}\pi r^2 h.$ Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
A garden fountain has a hemispherical basin with an inside radius of 20 cm. Water flows into the empty basin at a constant rate of 100 cm³ per second. When the water is $h
nbsp;cm deep, the volume of water in the basin is$ V = \frac{\pi}{3}h^2(60 - h)\text{ cm}^3. $
Find the rate at which the area of the water's circular surface is increasing at the instant the water is 8 cm deep.
You must use calculus and show any derivatives that you need to find when solving this problem.
A spherical weather balloon is filled with helium at a constant rate of 1.5 litres per second. Assume that the balloon is always a sphere, and that its volume is zero when the filling starts.
The volume and surface area of a sphere of radius $r$ are $V = \frac{4}{3}\pi r^3$ and $S = 4\pi r^2.$ Use 1 litre = 1000 cm³.
Find the rate at which the surface area of the balloon is increasing 20 seconds after the filling starts. Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
Hot-air balloon pilots steer by changing height, because the wind blows at different speeds at different heights.
Two balloons, A and B, are drifting in the same direction, both directly above a long, straight road across flat farmland, with B ahead of A. Balloon A is flying at a constant height of 650 m and drifting at a constant 6 m/s. Balloon B is flying at a constant height of 350 m.
When the direct distance between the balloons is 500 m, the angle of depression, $\theta,$ of balloon B from balloon A is increasing at a rate of 0.0024 radians per second.
Find the speed of balloon B at this instant.
You must use calculus and show any derivatives that you need to find when solving this problem.