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High School Math : How to find the radius of a sphere

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Example Question #2011 : High School Math

Find the radius of a sphere whose surface area is $SA=100cm^{2}$ $\displaystyle SA=100cm^{2}$.

Possible Answers:

2.82cm $\displaystyle 2.82cm$

Not enough information to solve

3.54cm $\displaystyle 3.54cm$

35.45cm $\displaystyle 35.45cm$

8.22cm $\displaystyle 8.22cm$

Correct answer:

2.82cm $\displaystyle 2.82cm$

Explanation:

We know that the surface area of the spere is $SA=100cm^{2}$ $\displaystyle SA=100cm^{2}$.

$SA=4\pi r^{2}$ $\displaystyle SA=4\pi r^{2}$

$100cm^{2}=4\pi r^{2}$ $\displaystyle 100cm^{2}=4\pi r^{2}$

Rearrange and solve for $\displaystyle r$.

$r^{2}=\frac{100cm^{2}}{4\pi}$ $\displaystyle r^{2}=\frac{100cm^{2}}{4\pi}$

$\dpi{100} r=\sqrt{\frac{100cm^{2}}{4\pi}}$ $\displaystyle \dpi{100} r=\sqrt{\frac{100cm^{2}}{4\pi}}$

$\dpi{100}\rightarrow 2.82cm$ $\displaystyle \dpi{100}\rightarrow 2.82cm$

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

What is the radius of a sphere that has a surface area of $16\pi$ $\displaystyle 16\pi$?

Possible Answers:

$\displaystyle 2$

$2\sqrt{\pi }$ $\displaystyle 2\sqrt{\pi }$

$8\sqrt{\pi }$ $\displaystyle 8\sqrt{\pi }$

$8\pi$ $\displaystyle 8\pi$

$\sqrt{2\pi }$ $\displaystyle \sqrt{2\pi }$

Correct answer:

$\displaystyle 2$

Explanation:

The standard equation to find the area of a sphere is $SA=4\pi r^2$ $\displaystyle SA=4\pi r^2$ where $\displaystyle r$ denotes the radius. Rearrange this equation in terms of $\displaystyle r$:

$r=\sqrt{\frac{SA}{4\pi }}$ $\displaystyle r=\sqrt{\frac{SA}{4\pi }}$

To find the answer, substitute the given surface area into this equation and solve for the radius:

$r=\sqrt{\frac{16\pi }{4\pi }}=\sqrt{4}=2$ $\displaystyle r=\sqrt{\frac{16\pi }{4\pi }}=\sqrt{4}=2$

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

Given that the volume of a sphere is $12\pi$ $\displaystyle 12\pi$, what is the radius?

Possible Answers:

$9\pi$ $\displaystyle 9\pi$

$\sqrt[3]{12 }$ $\displaystyle \sqrt[3]{12 }$

$\displaystyle 3$

$\sqrt[3]{9}$ $\displaystyle \sqrt[3]{9}$

$\sqrt{9\pi }$ $\displaystyle \sqrt{9\pi }$

Correct answer:

$\sqrt[3]{9}$ $\displaystyle \sqrt[3]{9}$

Explanation:

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}\pi r^3$ $\displaystyle V=\frac{4}{3}\pi r^3$ where $\displaystyle r$ denotes the radius. Rearrange this equation in terms of $\displaystyle r$:

$r=\sqrt[3]{\frac{3V}{4\pi }}$ $\displaystyle r=\sqrt[3]{\frac{3V}{4\pi }}$

Substitute the given volume into this equation and solve for the radius:

$r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 12\pi }{4\pi }}=\sqrt[3]{\frac{36\pi }{4\pi }}=\sqrt[3]{9 }$ $\displaystyle r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 12\pi }{4\pi }}=\sqrt[3]{\frac{36\pi }{4\pi }}=\sqrt[3]{9 }$

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Example Question #31 : Spheres

What is the radius of a sphere with a volume of $288\pi$ $\displaystyle 288\pi$?

Possible Answers:

$\displaystyle 12$

$\displaystyle 24$

$\displaystyle 36$

$\displaystyle 6$

Correct answer:

$\displaystyle 6$

Explanation:

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

$288\pi=\frac{4}{3}\pi r^3$ $\displaystyle 288\pi=\frac{4}{3}\pi r^3$

$\frac{288\pi}{\pi}=\frac{4\pi r^3}{3\pi}$ $\displaystyle \frac{288\pi}{\pi}=\frac{4\pi r^3}{3\pi}$

$288=\frac{4r^3}{3}$ $\displaystyle 288=\frac{4r^3}{3}$

$(288)(\frac{3}{4})=\frac{3}{4}(\frac{4r^3}{3})$ $\displaystyle (288)(\frac{3}{4})=\frac{3}{4}(\frac{4r^3}{3})$

216=r^3 $\displaystyle 216=r^3$

$\sqrt[3]{216}=\sqrt[3]{r^3}$ $\displaystyle \sqrt[3]{216}=\sqrt[3]{r^3}$

6=r $\displaystyle 6=r$

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High School Math : How to find the radius of a sphere

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2011 : High School Math

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

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

Example Question #31 : Spheres

All High School Math Resources

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