4. Applications of Derivatives

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.

a. Find an equation relating A to w.

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

a. Find an equation relating dV/dt to dr/dt.

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

b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?

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?