Question

The radius of a spherical balloon is increasing at the rate of 0.4 cm/minute. How fast is the volume changing when the radius is 7.8 cm? The volume is changing... (the rest of the text is not provided, but it seems like there's more to the question).

Answer 1

The volume is changing at a rate of approximately 307.98 cm³/minute when the radius is 7.8 cm.

Step-by-step explanation:

To find how fast the volume is changing, we can use the formula for the volume of a sphere:

V = (4/3)πr³

where V is the volume and r is the radius.

We know that the radius is increasing at a rate of 0.4 cm/minute. So, we can differentiate the volume equation with respect to time to find how the volume is changing with respect to time:

dV/dt = 4πr²(dr/dt)

Substituting the given values, we have:

dV/dt = 4π(7.8)²(0.4)

dV/dt ≈ 307.98 cm³/minute

Therefore, the volume is changing at a rate of approximately 307.98 cm³/minute when the radius is 7.8 cm.