air is being pumped into a spherical balloon at a rate of 8 cm^3/min. Find the rate at which the radius of the balloon is increasing when the balloon's radius is 5 cm.

2/25*3.14

To find the rate at which the radius of the balloon is increasing, we can use the related rates formula. Let's denote the radius of the balloon as r and the volume of the balloon as V.

Given the rate at which the volume is increasing, which is 8 cm^3/min, we want to find the rate at which the radius is increasing when the radius is 5 cm. This implies we need to determine dR/dt, the rate at which the radius is changing with respect to time.

First, we need to find an equation that relates the volume and the radius of the balloon. The formula for the volume of a sphere is V = (4/3)πr^3, where π is a constant (approximately 3.14159).

Now, take the derivative of both sides with respect to time t:

dV/dt = d/dt[(4/3)πr^3]

The left-hand side dV/dt represents the rate at which the volume is changing, which is given as 8 cm^3/min.

8 = d/dt[(4/3)πr^3]

To evaluate the right-hand side, we use the chain rule. Take the derivative of (4/3)πr^3 with respect to r, then multiply by dr/dt, the rate at which the radius is changing with respect to time.

8 = (4/3)π(3r^2)(dr/dt)

Simplifying:

8 = 4πr^2(dr/dt)

Now, rearrange the equation to solve for dr/dt:

dr/dt = 8 / [4πr^2]

Plug in the value for r = 5 cm:

dr/dt = 8 / [4π(5^2)]

dr/dt = 8 / [4π(25)]

dr/dt = 8 / [100π]

So, the rate at which the radius of the balloon is increasing when the radius is 5 cm is 8 / [100π] cm/min.