Sphere Volume and Radius Calculation

This problem asks us to find the radius of a sphere when we know its volume. We are given the volume and the value of pi ($\pi$) to use in our calculations.

Given Information

• Volume of the sphere, $V = 38808 \text{ cm}^3$
• Value of pi, $\pi = \frac{22}{7}$

Sphere Volume Formula

The formula for the volume ($V$) of a sphere with radius ($r$) is:

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

Finding the Radius Step-by-Step

Our goal is to find the radius ($r$). We need to rearrange the formula to solve for $r$.

1. Start with the volume formula: $$V = \frac{4}{3} \pi r^3$$
2. Isolate $r^3$: To get $r^3$ by itself, we can multiply both sides by 3 and divide by $4\pi$. $$r^3 = \frac{3V}{4\pi}$$
3. Substitute the given values: Now, plug in the values for $V$ and $\pi$. $$r^3 = \frac{3 \times 38808 \text{ cm}^3}{4 \times \frac{22}{7}}$$
4. Simplify the denominator: Calculate $4 \times \frac{22}{7}$. $$4 \times \frac{22}{7} = \frac{88}{7}$$
5. Substitute the simplified denominator back into the equation: $$r^3 = \frac{3 \times 38808}{\frac{88}{7}}$$
6. Simplify the fraction: Dividing by a fraction is the same as multiplying by its reciprocal. $$r^3 = (3 \times 38808) \times \frac{7}{88}$$ $$r^3 = \frac{3 \times 38808 \times 7}{88}$$
7. Perform the division: Let's divide 38808 by 88. $$38808 \div 88 = 441$$
8. Continue simplifying: $$r^3 = 3 \times 441 \times 7$$
9. Calculate the final value of $r^3$: $$r^3 = 3 \times 3087$$ $$r^3 = 9261$$
10. Find the radius ($r$): To find $r$, we need to take the cube root of 9261. $$r = \sqrt[3]{9261}$$ We know that $20^3 = 8000$ and $30^3 = 27000$, so the number should be between 20 and 30. Let's check $21^3$: $$21 \times 21 \times 21 = 441 \times 21 = 9261$$ So, $r = 21$.
11. Include the units: Since the volume was in cubic centimeters ($\text{cm}^3$), the radius will be in centimeters (cm). $$r = 21 \text{ cm}$$

Conclusion

By using the formula for the volume of a sphere and performing algebraic manipulations and calculations, we found the radius of the sphere to be 21 cm.