The height of a cylinder with a fixed radius of 10 cm is increasing at the rate of 0.5 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 30cm.
To find the rate of change of the volume of the cylinder, we can use the formula for the volume of a cylinder:
V = πr^2h
where V is the volume, r is the radius, and h is the height.
We are given that the radius, r, is fixed at 10 cm, and the height, h, is increasing at a rate of 0.5 cm/min. We need to find the rate of change of the volume, dV/dt, when the height is 30 cm.
First, we differentiate the volume formula with respect to time, t:
dV/dt = d/dt (πr^2h)
To find dV/dt, we need to find the values of r, h, and dh/dt (rate of change of height) when h = 30 cm.
Given:
r = 10 cm,
h = 30 cm,
dh/dt = 0.5 cm/min.
Substituting these values into the formula, we have:
dV/dt = d/dt (π(10^2)(30))
To simplify, we can calculate (10^2) before differentiating:
dV/dt = d/dt (π(100)(30))
dV/dt = d/dt (3000π)
Since the radius, r, is fixed, its derivative is zero. Therefore, we have:
dV/dt = 0 + (3000π)(0)
dV/dt = 0
Thus, the rate of change of the volume of the cylinder is 0 cm^3/min when the height is 30 cm.
To find the rate of change of the volume of the cylinder, we need to calculate the derivative of the volume function with respect to time.
The volume of a cylinder is given by the formula:
V = πr^2h
Where:
V is the volume,
r is the radius, and
h is the height.
Taking the derivative of the volume function with respect to time (t), using the chain rule, we get:
dV/dt = dV/dh * dh/dt
We are given that the radius (r) is fixed at 10 cm, and the rate of change of the height (dh/dt) is 0.5 cm/min.
Now, let's calculate the derivative of the volume function, dV/dh:
dV/dh = 2πrh
Plugging in the given values:
dV/dh = 2π(10)(30)
= 600π
Now, we can calculate the rate of change of the volume:
dV/dt = dV/dh * dh/dt
= (600π) * (0.5)
= 300π
Therefore, the rate of change of the volume of the cylinder when the height is 30 cm is 300π cubic centimeters per minute.
v = pi r^2 h = 100pi h
dv/dt = 100pi dh/dt
Now just plug in your numbers to get dv/dt.
Note that it does not matter what the height is. Since the cylinder has a constant radius, its volume is directly proportional to its height.