The height of a cylinder with a fixed radius of 4 cm is increasing at the rate of 2 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 14cm.
Well, let me do some circus calculations for you! The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
We are given that the radius r is fixed at 4 cm. So, in this case, the volume equation becomes V = π(4^2)h, which simplifies to V = 16πh.
Now, we need to find the rate of change of the volume with respect to time. To do this, we will differentiate the equation with respect to time:
dV/dt = d/dt (16πh)
Now, since h is changing with respect to time, we need to use the chain rule. The chain rule states that if y = f(g(t)), then dy/dt = dy/dg * dg/dt.
Let's differentiate the equation again:
dV/dt = d(16π)/dt * dh/dt
The first part of the equation, d(16π)/dt, is simply 0 since 16π is a constant. So, we are left with:
dV/dt = 0 * dh/dt
Therefore, the rate of change of the volume is 0 when the height is 14 cm. It seems like this cylinder is playing tricks on us and remaining constant in volume while increasing in height!
To find the rate of change of the volume of the cylinder with respect to time, we need to use the formula for the volume of a cylinder.
The volume of a cylinder can be calculated using the formula: V = πr^2h, where V is the volume, r is the radius, and h is the height.
In this case, the radius (r) is fixed at 4 cm. Therefore, we only need to consider the rate of change of the height (h), which is given as 2 cm/min.
To find the rate of change of the volume (dV/dt) when the height is 14cm, we first differentiate the volume formula with respect to time (t):
dV/dt = π(2rh)dh/dt
Substituting the given values:
r = 4 cm
h = 14 cm
dh/dt = 2 cm/min
Plug in these values into the formula:
dV/dt = π(2(4)(14))(2)
= 4π(4)(2)
= 32π cm^3/min
Therefore, the rate of change of the volume of the cylinder when the height is 14 cm is 32π cm^3/min.
To find the rate of change of the volume of the cylinder, we need to use the formula for the volume of a cylinder, which is given by V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given:
Radius (r) = 4 cm
Rate of change of height (dh/dt) = 2 cm/min
Height (h) = 14 cm
To find the rate of change of volume (dV/dt), we need to differentiate the volume formula with respect to time (t):
dV/dt = d/dt (πr^2h)
Now, let's substitute the given values:
dV/dt = d/dt (π(4^2)(14))
To simplify, we will use the properties of differentiation. Since the radius (r) is constant, its derivative with respect to time is zero.
dV/dt = π(4^2) * d/dt (14)
Since the derivative of a constant is zero, we can further simplify:
dV/dt = π(16) * 0
Therefore, the rate of change of the volume of the cylinder (dV/dt) is 0 cm^3/min.
The rate of change of the volume will be zero because in this case, the height of the cylinder is increasing while the radius is fixed. When the radius is not changing, the volume of the cylinder will not change over time.
At a time of t minutes, let the height be h cm
V = π r^2 h = 16π h
dV/dt = 16π dh/dt
but dh/dt = 2 cm/min , and the height is not an issue, since the variable h does not appear in dV/dt
dV/dt = 16π(2) = 32π cm^3/min