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Mindy is a sculptor. She has a cylinder of stone with a radius of $3\,\text{meters}\,\frac{v}{3\pi}$ and a height of $2\,\text{meters}$ . She needs to carve out a sphere of radius $1\,\text{meter}$ from the cylinder. Mindy must cut away $\text{cubic meters}$ of stone from the cylinder in order to be left with the sphere. What is the value of $v$ ?

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Solve proportions: word problems

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Q. Mindy is a sculptor. She has a cylinder of stone with a radius of

3\,\text{meters}\,\frac{v}{3\pi}

and a height of

2\,\text{meters}

. She needs to carve out a sphere of radius

1\,\text{meter}

from the cylinder. Mindy must cut away

\text{cubic meters}

of stone from the cylinder in order to be left with the sphere. What is the value of

v

?

Calculate Cylinder Volume: Calculate the volume of the cylinder: $\newline$ Volume of cylinder = $\pi \times \text{radius}^2 \times \text{height}$ $\newline$ = $\pi \times \left(\frac{3}{\pi}\right)^2 \times 2$ $\newline$ = $\pi \times \left(\frac{9}{\pi^2}\right) \times 2$ $\newline$ = $\left(\frac{18}{\pi}\right)$ cubic meters
Calculate Sphere Volume: Calculate the volume of the sphere: $\newline$ Volume of sphere = $(\frac{4}{3}) \pi \text{radius}^3$ $\newline$ = $(\frac{4}{3}) \pi 1^3$ $\newline$ = $(\frac{4}{3}) \pi$ cubic meters
Calculate Volume to Cut: Calculate the volume of stone Mindy needs to cut away: $\newline$ Volume to cut away = Volume of cylinder - Volume of sphere $\newline$ = $(\frac{18}{\pi}) - (\frac{4}{3}) \times \pi$ $\newline$ = $(\frac{18}{\pi}) - (\frac{4\pi}{3})$ $\newline$ = $\frac{54 - 4\pi^2}{3\pi}$ cubic meters

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