The height of a cylinder with a fixed radius of 6 cm is increasing at the rate of 3 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 20cm.
v = pi r^2 h
dv/dt = pi r^2 dh/dt
To find the rate of change of the volume of the cylinder with respect to time, 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.
Given:
- Radius (r) is fixed at 6 cm.
- The height (h) is increasing at a rate of 3 cm/min.
We need to find the rate of change of the volume (dV/dt) when the height (h) is 20 cm.
First, let's differentiate the volume equation with respect to time (t):
dV/dt = d/dt (πr^2h)
Now, let's substitute the known values:
V = π(6^2)(20)
dV/dt = d/dt (π(6^2)(20))
To differentiate, we need to apply the product rule. Differentiating the volume equation:
dV/dt = π * (2r) * (dh/dt) + π * (r^2) * (dh/dt)
Substituting known values:
dV/dt = π * (2 * 6) * (3) + π * (6^2) * (3)
Simplifying:
dV/dt = 36π + 108π
dV/dt = 144π
Therefore, the rate of change of the volume of the cylinder with respect to time, when the height is 20 cm, is 144π cubic cm/min.
To find the rate of change of the volume of the cylinder with respect to time, we need to differentiate the volume formula with respect to time and then substitute the given values to calculate the rate.
First, let's recall the formula for the volume of a cylinder:
V = πr^2h
Where V represents the volume, r is the radius, and h is the height.
To differentiate the volume formula with respect to time, we can use the chain rule of differentiation. The chain rule states that if we have a function that is the composition of two functions, then the derivative of that composite function is the derivative of the outer function multiplied by the derivative of the inner function.
In this case, the volume V is a function of both the radius r and the height h, which are both functions of time (t). Therefore, we can write the volume formula as a composition of functions:
V(t) = π[r(t)]^2 * [h(t)]
To differentiate this formula with respect to time, we apply the chain rule.
dV/dt = dV/dr * dr/dt + dV/dh * dh/dt
Since we are interested in the rate of change of the volume with respect to time when the height is 20 cm, we substitute the given values:
r = 6 cm (fixed radius)
dh/dt = 3 cm/min (rate of change of height)
h = 20 cm (height at which we want to find the rate)
Now, let's differentiate the volume formula:
dV/dr = d(π[r(t)]^2)/dr = 2πr
dV/dh = d(π[r(t)]^2 * h)/dh = π[r(t)]^2
Substituting the given values:
dV/dr = 2π(6) = 12π
dV/dh = π(6)^2 = 36π
Now, let's calculate the rate of change of the volume with respect to time using the formula:
dV/dt = dV/dr * dr/dt + dV/dh * dh/dt
dV/dt = 12π * 0 + 36π * 3
= 0 + 108π
= 108π
Therefore, the rate of change of the volume of the cylinder when the height is 20 cm is 108π cubic cm per minute.