A spherical balloon is inflated with helium at the rate of 100pi ft ^3/min. How fast is the ballon's radius increasing at the instant the radius is 5ft.?

To find how fast the balloon's radius is increasing, we can use the relationship between the volume and the radius of a spherical balloon.

The volume of a sphere is given by the formula:

V = (4/3)πr^3

where V is the volume and r is the radius.

We are given that the balloon is being inflated at a rate of 100π ft^3/min, so the rate of change of volume (dV/dt) is 100π ft^3/min.

We need to find the rate at which the radius is changing (dr/dt) when the radius is 5 ft.

To do this, we can differentiate the volume formula with respect to time (t):

dV/dt = d/dt((4/3)πr^3)

Now, let's differentiate each term on the right side of the equation:

dV/dt = (4/3)π * d/dt(r^3)

Using the chain rule, the derivative of r^3 with respect to time is:

3r^2 * dr/dt

Substituting back into the original equation:

100π = (4/3)π * 3r^2 * dr/dt

Simplifying:

100 = 4r^2 * dr/dt

Now, let's solve for dr/dt by dividing both sides by 4r^2:

(4r^2 * dr/dt) / 4r^2 = 100 / 4r^2

dr/dt = 25 / r^2

Substituting the given radius of 5 ft:

dr/dt = 25 / (5^2)

dr/dt = 25 / 25

dr/dt = 1 ft/min

Therefore, the balloon's radius is increasing at a rate of 1 ft/min when the radius is 5 ft.

To find the rate at which the balloon's radius is increasing, we can use the formula for the volume of a sphere:

V = (4/3)πr^3

Where V is the volume and r is the radius.

We are given that the balloon is being inflated at a rate of 100π ft^3/min, so we can express the rate of change of the volume as dV/dt = 100π.

To find the rate at which the radius is increasing at a specific instant, we need to find dr/dt.

Differentiating both sides of the volume equation with respect to time t:

dV/dt = 4πr^2 * dr/dt

Substituting the given values, we have:

100π = 4π(5^2) * dr/dt

Simplifying the equation:

100 = 4(25) * dr/dt

100 = 100 * dr/dt

dr/dt = 1 ft/min

Therefore, the balloon's radius is increasing at a rate of 1 ft/min when the radius is 5 ft.

v = 4/3 pi r^3

dv/dt = 4 pi r^2 dr/dt

100 = 4 pi * 25 dr/dt
dr/dt = 1/pi