A water tank in the shape of a right circular cylinder has a volume that is increasing at the rate
of 300pi cubic centimeters per hour. If the height is decreasing at the rate of 15 centimeters per hour,
find the rate at which the radius is changing when the radius is 20
centimeters and the height is 30 centimeters.
v = 1/3 pi r^2 h
dv/dt = pi/3 (2rh dr/dt + r^2 dh/dt)
Now just plug in
dv/dt = 300pi
dh/dt = -15
r=20
h=30
and solve for dr/dt
To find the rate at which the radius is changing, we can use the formula for the volume of a cylinder:
V = πr^2h
Where V is the volume of the cylinder, r is the radius, and h is the height.
Given that the volume is increasing at a rate of 300π cubic centimeters per hour, we can express this as:
dV/dt = 300π
Given that the height is decreasing at a rate of 15 centimeters per hour, we can express this as:
dh/dt = -15
To find the rate at which the radius is changing (dr/dt), we need to use the chain rule of differentiation. The chain rule states that:
dV/dt = dV/dr * dr/dt + dV/dh * dh/dt
Since we want to find dr/dt, we can rearrange the equation as follows:
dV/dr * dr/dt = dV/dt - dV/dh * dh/dt
Now, let's differentiate the equation of the volume with respect to r:
dV/dr = d(πr^2h)/dr = 2πrh
Now, substitute the given values into the equations:
dV/dr = 2π(20)(30) = 1200π
dV/dh = πr^2 = π(20)^2 = 400π
dV/dt = 300π
dh/dt = -15
Substituting the values into the equation we rearranged earlier:
(1200π) * dr/dt = 300π - (400π)(-15)
Simplifying, we get:
(1200π) * dr/dt = 4500π
Now, divide both sides by 1200π:
dr/dt = (4500π) / (1200π) = 3.75 cm/h
Therefore, the rate at which the radius is changing when the radius is 20 centimeters and the height is 30 centimeters is 3.75 centimeters per hour.