Question

a spherical balloon is inflated so that its volume is increasing at the rate of 3.2 ft³/min . how rapidly is the diameter of the balloon increasing when the diameter is 1.2 feet? the diameter is increasing at 100/(9pi) ft/min.

Answer 1

To find the rate at which the diameter of the balloon is increasing, use the relationship between the volume of a sphere and its diameter. Differentiate both sides of the equation concerning time to find the rate of change of volume concerning time. Substitute the given values into the equation to solve for the rate at which the diameter is changing.

Step-by-step explanation:

To find the rate at which the diameter of the balloon is increasing, we will use the relationship between the volume of a sphere and its diameter. The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. Since the diameter is twice the radius, we can rewrite the formula as V = (4/3)π(d/2)³, where d is the diameter.

Now, we can differentiate both sides of the equation concerning time (t) to find the rate of change of volume concerning time. We have dV/dt = (4/3)π(3/2)(d/2)²(d'/dt), where d' is the rate at which the diameter is changing.

Given that dV/dt = 3.2 ft³/min, we can substitute this value into the equation and solve for d'/dt:

3.2 = (4/3)π(3/2)(1.2/2)²(d'/dt)

d'/dt = (3.2 ft³/min) / ((4/3)π(3/2)(1.2/2)²) ≈ 100/(9π) ft/min