The diameter and height of a right circular cylinder are found at a certain instant to be 10 and 20 inches respectively. If the diameter is increasing at the rate of 1 inch per minute, what change in the height will keep the colume constant?
The diameter and height of a right circular cylinder are found at a certain instant to be 10 and 20 inches respectively. If the diameter is increasing at the rate of 1 inch per minute, what change in the height will keep the colume constant?
$\displaystyle V = Ah = \pi r^2h = \pi(d/2)^2h$
$\displaystyle \frac{dV}{dt} = \pi(d/2)\frac{dd}{dt}h + \pi(d/2)^2\frac{dh}{dt}$
D/2 = r = 5, h = 10
$\displaystyle \frac{dV}{dt} = 100\pi\frac{dd}{dt} + 25\pi\frac{dh}{dt}$
$\displaystyle \frac{dd}{dt} = 1$
$\displaystyle \frac{dV}{dt} = 100\pi + 25\pi\frac{dh}{dt}$
If volume is contant, dV/dt = 0
$\displaystyle 0 = 100\pi + 25\pi\frac{dh}{dt}$