Air is being pumped into a spherical balloon at a rate of 2 cubic meters per minute. At what rate is the radius changing when the volume of the balloon is 36pi cubic meters?
V = (4/3)π r^3
when V = 36π
36π = (4/3)π r^3
27 = r^3
r = 3 m
dV/dt = 4π r^2 dr/dt
2 = 4π (9)dr/dt
dr/dt = 2/(36π) m/min
= appr .01768 m/min
To find the rate at which the radius is changing, we need to use the relationship between the radius and the volume of a sphere.
The formula for the volume V of a sphere with radius r is given by:
V = (4/3) * π * r^3
Let's differentiate both sides of the equation with respect to time (t) to find the rate of change.
dV/dt = d((4/3) * π * r^3)/dt
Now, let's differentiate each term separately using the power rule:
dV/dt = (4/3) * π * d(r^3)/dt
Next, we can simplify the right-hand side by applying the chain rule:
dV/dt = (4/3) * π * 3r^2 * dr/dt
Since we want to find the rate at which the radius is changing (d(r)/dt), we can rearrange the equation to solve for (dr/dt):
dr/dt = (dV/dt) / [(4/3) * π * 3r^2]
Given that the volume is changing at a rate of 2 cubic meters per minute (dV/dt = 2), and the volume is 36π cubic meters, we can substitute these values into the equation:
dr/dt = 2 / [(4/3) * π * 3r^2]
Now, we can substitute the volume V = 36π into the volume formula to find the radius r:
36π = (4/3) * π * r^3
Dividing both sides by (4/3) * π, we get:
3r^3 = 36
Simplifying further:
r^3 = 12
Taking the cube root of both sides:
r = ∛12 = 2∛3
Now, substitute this value for r into the equation for dr/dt:
dr/dt = 2 / [(4/3) * π * 3(2∛3)^2]
dr/dt = 2 / [(4/3) * π * 12∛3]
dr/dt = 2 / [(16/3) * π * ∛3]
Simplifying:
dr/dt = 6 / [16π *∛3]
The rate at which the radius is changing when the volume of the balloon is 36π cubic meters is approximately equal to 6 / [16π * ∛3] cubic meters per minute.