The radius of a spherical balloon is increasing at a rate of 2 centimeters per minute. How fast is the volume changing when the radius is 8 centimeters?
Note: The volume of a sphere is given by 4(pi)r^3
So, if V = 4/3 pi r^3
dV = 4 pi * r^2 dr
dV = 4 pi * 8^2 * 2
= 512 pi cc/min
To find how fast the volume is changing when the radius is 8 centimeters, we can use the chain rule from calculus.
Let's denote the volume as V and the radius as r. We are given that the radius is increasing at a rate of 2 centimeters per minute, so we have dr/dt = 2 cm/min.
We are asked to find dV/dt, the rate at which the volume is changing when the radius is 8 centimeters.
The formula for the volume of a sphere is V = 4/3 * pi * r^3. We can take the derivative of both sides with respect to time t:
dV/dt = d/dt (4/3 * pi * r^3)
Applying the chain rule, we get:
dV/dt = (dV/dr) * (dr/dt)
The first term, dV/dr, is the derivative of the volume with respect to the radius. We can find this by taking the derivative of the volume formula:
dV/dr = d/dt (4/3 * pi * r^3) = 4 * pi * r^2
Now we can substitute the given values: r = 8 centimeters and dr/dt = 2 cm/min:
dV/dt = (4 * pi * r^2) * (dr/dt) = (4 * pi * 8^2) * 2
Simplifying further:
dV/dt = (4 * pi * 64) * 2 = 512 * pi cm^3/min
So, when the radius is 8 centimeters, the volume of the balloon is changing at a rate of 512 * pi cubic centimeters per minute.
To find how fast the volume of the balloon is changing, we need to differentiate the volume equation with respect to time.
Given:
The rate of change of the radius (dr/dt) = 2 cm/min.
The radius of the balloon (r) = 8 cm.
The formula for the volume of a sphere is:
V = (4/3)πr^3
Differentiate both sides of the equation with respect to time (t):
dV/dt = d/dt [(4/3)πr^3]
Using the chain rule, the derivative of r^3 with respect to t is:
d(r^3)/dt = 3r^2 * (dr/dt)
Substituting the given values, we have:
dV/dt = (4/3)π * 3(8)^2 * 2
Simplifying this expression:
dV/dt = (4/3)π * 3 * 64 * 2
dV/dt = 512π cm^3/min
Therefore, the volume of the balloon is changing at a rate of 512π cm^3/min when the radius is 8 cm.