4. Applications of Derivatives

The volume V of a sphere of radius r changes over time t.

c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Shrinking square The sides of a square decrease in length at a rate of 1 m/s.

a. At what rate is the area of the square changing when the sides are 5 m long?

The sides of a square decrease in length at a rate of 1 m/s.

b. At what rate are the lengths of the diagonals of the square changing?

The legs of an isosceles right triangle increase in length at a rate of 2 m/s.

c. At what rate is the length of the hypotenuse changing?

Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.

a. At what rate is the area of the triangle changing when the legs are 5 m long?

The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?

Once Kate’s kite reaches a height of 50 ft (above her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kate’s hands at the moment when she has released 120 ft of string?

A bug is moving along the right side of the parabola y=x² at a rate such that its distance from the origin is increasing at 1 cm/min.

b. Use the equation y=x² to find an equation relating dy/dt to dx/dt.

A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft³/min) if the water level is dropping at 6 min/in?

Water is drained out of an inverted cone that has the same dimensions as the cone depicted in Exercise 36. If the water level drops at 1 ft/min, at what rate is water (in ft³/min) draining from the tank when the water depth is 6 ft? 

A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?