Question

The volume of a soap bubble is d
2,756.9 mm^3. Find the radius and diameter of the soap bubble. Use 3.14 for piπ. Question content area bottom Part 1 The radius of the bubble is about 8.78.7 mm. ​(Round to the nearest tenth as​ needed.) Part 2 The diameter of the bubble is about enter your response her

Answer 1

To find the radius and diameter of a soap bubble with a volume of \(2,756.9 \, \text{mm}^3\), we can use the formula for the volume of a sphere:

\[
V = \frac{4}{3} \pi r^3
\]

where \(V\) is the volume and \(r\) is the radius. We can solve for \(r\) using the given volume and \( \pi \) approximated as \(3.14\).

1. **Insert the volume into the formula:**
\[
2,756.9 = \frac{4}{3} \cdot 3.14 \cdot r^3
\]

2. **Rearrange to find \(r^3\):**
\[
r^3 = \frac{2,756.9 \cdot 3}{4 \cdot 3.14}
\]

3. **Calculate the right side:**
\[
r^3 = \frac{2,756.9 \cdot 3}{12.56} \approx \frac{8,270.7}{12.56} \approx 658.31
\]

4. **Take the cube root to find \(r\):**
\[
r \approx \sqrt[3]{658.31} \approx 8.68 \, \text{mm}
\]

Rounding to the nearest tenth:
\[
r \approx 8.7 \, \text{mm}
\]

5. **Calculate the diameter \(d\):**
The diameter is twice the radius:
\[
d = 2r \approx 2 \cdot 8.7 \approx 17.4 \, \text{mm}
\]

### Final Answers:
- The radius of the bubble is about \(8.7 \, \text{mm}\).
- The diameter of the bubble is about \(17.4 \, \text{mm}\).