A sphere has a volume of $36\pi$ $in^3$. Find the radius of the sphere. Use 3.14 for pi.
Understand the Problem
The question asks us to find the radius of a sphere, given its volume and using 3.14 as an approximation for pi. We will use the formula for the volume of a sphere to solve for the radius.
Answer
The radius of the sphere is $r=3$ inches.
Answer for screen readers
The radius of the sphere is 3 inches.
Steps to Solve
- Write the formula for the volume of a sphere
The volume $V$ of a sphere is given by the formula: $$V = \frac{4}{3}\pi r^3$$ where $r$ is the radius of the sphere.
- Substitute the given volume into the formula
We are given that the volume $V = 36\pi$ in$^3$. Substituting this into the formula gives us: $$36\pi = \frac{4}{3}\pi r^3$$
- Solve for $r^3$
First, we can divide both sides of the equation by $\pi$: $$36 = \frac{4}{3}r^3$$ Next, multiply both sides by $\frac{3}{4}$ to isolate $r^3$: $$36 \cdot \frac{3}{4} = r^3$$ $$27 = r^3$$
- Solve for $r$
To find the radius $r$, we take the cube root of both sides: $$r = \sqrt[3]{27}$$ $$r = 3$$
The radius of the sphere is 3 inches.
More Information
The volume was given in terms of pi, which simplified the calculation. The problem specifies to use 3.14 for pi, but using $\pi$ allows for finding the exact answer.
Tips
A common mistake is forgetting the formula for the volume of a sphere. Another mistake could be made in the algebraic manipulation to isolate $r$. The formula is $V = \frac{4}{3} \pi r^3$. Make sure to multiply by $\frac{3}{4}$ and not $\frac{4}{3}$ when isolating $r^3$. Another common mistake is taking the square root instead of the cube root.
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