Question

The volume of a sphere is 500
3
πcm3. What is the radius?

Sphere V = 4
3
πr3

1. Substitute value into formula:  500
3
π = 4
3
πr3

2. Undo multiplication of 4
3
:    125π = πr3

3. Undo multiplication of π:    125 = r3

4. Undo cube:         3√125 = r

The radius of the sphere is

Answer 1

To find the radius of the sphere, we can follow the steps you outlined:

1. **Substitute the value into the formula**:
\[
V = \frac{4}{3} \pi r^3
\]
Given \( V = 500 \frac{3}{\pi} \), so:
\[
500 \frac{3}{\pi} = \frac{4}{3} \pi r^3
\]

2. **Multiply both sides by \( \frac{3}{\pi} \)**:
\[
500 = \frac{4}{3} r^3
\]

3. **Multiply both sides by \( \frac{3}{4} \)** to undo the multiplication of \( \frac{4}{3} \):
\[
r^3 = \frac{3 \times 500}{4} = \frac{1500}{4} = 375
\]

4. **Now, to find \( r \)**, take the cube root:
\[
r = \sqrt[3]{375}
\]

Now, let's simplify \( \sqrt[3]{375} \):

375 can be factored as \( 375 = 3 \times 125 = 3 \times 5^3 \). Therefore:
\[
r = \sqrt[3]{3 \times 5^3} = 5 \sqrt[3]{3}
\]

Thus, the radius of the sphere is \( 5 \sqrt[3]{3} \) cm. If you want a decimal approximation, you can calculate \( \sqrt[3]{3} \approx 1.442 \), yielding:
\[
r \approx 5 \times 1.442 \approx 7.21 \text{ cm}
\]

So, the radius of the sphere is approximately \( 7.21 \) cm, or exactly \( 5 \sqrt[3]{3} \) cm.