Question

the height of a cylinder is increasing at a constant rate of 2 centimeters per second, and the volume is increasing at a rate of 1152 cubic centimeters per second. at the instant when the height of the cylinder is 33 centimeters and the volume is 403 cubic centimeters, what is the rate of change of the radius? the volume of a cylinder can be found with the equation v

Answer 1

about 2.758 cm/s

Explanation:

You want the rate of change of a cylinder's radius, when its height is 33 cm increasing at 2 cm/s, and its volume is 403 cm³ increasing at 1152 cm³/s.

Rate of change

The volume of a cylinder is given by the equation ...

V = πr²h

Solving for the radius gives ...

r = √(V/(πh))

Differentiating, we have ...

`r'=(1)/(√(\pi))\left((1)/(2)V^{-(1)/(2)}h^{-(1)/(2)}V'-(1)/(2)V^{(1)/(2)}h^{-(3)/(2)}h'\right)=(hV'-Vh')/(2h√(\pi Vh))`

Filling in the values V = 403, V' = 1152, h = 33, h' = 2, we have ...

r' = (33·1152 -403·2)/(2·33·√(π·403·33)) = 37210/(66√(13299π))

r' ≈ 2.758 . . . . . cm/s

The rate of change of the radius is about 2.758 cm/s.