A spherical balloon is inflated and its volume increases at a rate of . What is the rate of change of its radius when the radius is 9 in.?
The balloon’s radius is increasing at a rate of
(Round to three decimal places as needed.)

Ask by Chan Gibbs. in the United States

Mar 12,2025

Question

A spherical balloon is inflated and its volume increases at a rate of . What is the rate of change of its radius when the radius is 9 in.?
The balloon’s radius is increasing at a rate of
(Round to three decimal places as needed.)

Ask by Chan Gibbs. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

We start with the formula for the volume of a sphere:

$$
V=\frac{4}{3}\pi r^3
$$

Differentiate both sides with respect to time $$ t $$:

$$
\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}
$$

We are given that $$\frac{dV}{dt}=23$$ in$$^3$$/min and the radius $$r=9$$ in, so substitute these values into the equation:

$$
23 = 4\pi (9)^2 \frac{dr}{dt}
$$

Simplify the equation:

$$
23 = 4\pi (81)\frac{dr}{dt} = 324\pi \frac{dr}{dt}
$$

Now solve for $$\frac{dr}{dt}$$:

$$
\frac{dr}{dt}=\frac{23}{324\pi}
$$

Calculating the numerical value:

$$
\frac{dr}{dt} \approx \frac{23}{324 \times 3.14159} \approx \frac{23}{1017.876} \approx 0.0226\ \text{in/min}
$$

Rounded to three decimal places, the rate of change of the balloon's radius is:

$$
\frac{dr}{dt}\approx 0.023\ \text{in/min}
$$
The balloon's radius is increasing at a rate of approximately 0.023 inches per minute.