A spherical balloon is designed to pop at the instant its radius has reached 8 centimeters.
Assuming the balloon is filled with helium at a rate of 10 cubic centimeters per second,
calculate how fast the radius is growing at the instant it pops. (Round your answer to
two decimal places.) (The volume of a sphere with radius is given by .)

Ask by Frazier Mitchell. in the United States

Jan 25,2025

Question

A spherical balloon is designed to pop at the instant its radius has reached 8 centimeters.
Assuming the balloon is filled with helium at a rate of 10 cubic centimeters per second,
calculate how fast the radius is growing at the instant it pops. (Round your answer to
two decimal places.) (The volume of a sphere with radius is given by .)

Ask by Frazier Mitchell. in the United States

Jan 25,2025

UpStudy AI · Accepted Answer

To determine how fast the radius of the balloon is growing at the instant it pops, we'll use the relationship between the volume of a sphere and its radius.

**Given:**
- Volume of a sphere: $$ V = \frac{4}{3} \pi r^3 $$
- Rate at which helium is added: $$ \frac{dV}{dt} = 10 $$ cm³/s
- Radius at the instant of popping: $$ r = 8 $$ cm

**Steps:**

1. **Differentiate the volume with respect to time:**
   $$
   \frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \frac{dr}{dt}
   $$
   
2. **Plug in the known values:**
   $$
   10 = 4 \pi (8)^2 \frac{dr}{dt}
   $$
   $$
   10 = 4 \pi \times 64 \frac{dr}{dt}
   $$
   $$
   10 = 256 \pi \frac{dr}{dt}
   $$

3. **Solve for $$\frac{dr}{dt}$$:**
   $$
   \frac{dr}{dt} = \frac{10}{256 \pi} = \frac{5}{128 \pi} \text{ cm/s}
   $$
   
4. **Calculate the numerical value:**
   $$
   \frac{5}{128 \pi} \approx \frac{5}{402.123} \approx 0.01243 \text{ cm/s}
   $$
   
5. **Round to two decimal places:**
   $$
   \frac{dr}{dt} \approx 0.01 \text{ cm/s}
   $$

**Answer:**  
At the instant the balloon pops, the radius is increasing at approximately 0.01 cm per second.
At the instant the balloon pops, the radius is increasing at approximately 0.01 cm per second.

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