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ISEE Upper Level Quantitative : How to find the radius of a sphere

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Example Questions

Example Question #1 : How To Find The Radius Of A Sphere

The volume of a sphere is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ \sqrt{3 \pi } } {2 \pi} \textrm{ in}$ $\displaystyle \frac{ \sqrt{3 \pi } } {2 \pi} \textrm{ in}$

$\frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$ $\displaystyle \frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

$\frac{ 6\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$ $\displaystyle \frac{ 6\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

$\frac{ 18\sqrt{3 \pi } } {\pi} \textrm{ in}$ $\displaystyle \frac{ 18\sqrt{3 \pi } } {\pi} \textrm{ in}$

$\frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}\textrm{ in}$ $\displaystyle \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}\textrm{ in}$

Correct answer:

$\frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$ $\displaystyle \frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

Explanation:

The volume $\displaystyle V$ of a sphere with radius $\displaystyle r$ is

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

To find the radius in yards, we set V = 1 $\displaystyle V = 1$ and solve for $\displaystyle r$.

$1 = \frac{4}{3} \pi r^{3}$ $\displaystyle 1 = \frac{4}{3} \pi r^{3}$

$\frac{3}{4} \cdot 1 =\frac{3}{4} \cdot \frac{4}{3} \pi r^{3}$ $\displaystyle \frac{3}{4} \cdot 1 =\frac{3}{4} \cdot \frac{4}{3} \pi r^{3}$

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

$\frac{1}{\pi} \cdot \frac{3}{4} =\frac{1}{\pi} \cdot \pi r^{3}$ $\displaystyle \frac{1}{\pi} \cdot \frac{3}{4} =\frac{1}{\pi} \cdot \pi r^{3}$

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

$r = \sqrt[3]{ \frac{3}{4\pi} } = \sqrt[3]{ \frac{3\cdot 2 \pi^{2}}{4\pi \cdot 2 \pi^{2}} }= \sqrt[3]{ \frac{6 \pi^{2}}{8 \pi^{3}} } = \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}$ $\displaystyle r = \sqrt[3]{ \frac{3}{4\pi} } = \sqrt[3]{ \frac{3\cdot 2 \pi^{2}}{4\pi \cdot 2 \pi^{2}} }= \sqrt[3]{ \frac{6 \pi^{2}}{8 \pi^{3}} } = \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}$ yards.

Since the problem requests the radius in inches, multiply by 36:

$36 \times \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi} = \frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi}$ $\displaystyle 36 \times \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi} = \frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi}$

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ISEE Upper Level Quantitative : How to find the radius of a sphere

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

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Example Question #1 : How To Find The Radius Of A Sphere

All ISEE Upper Level Quantitative Resources

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