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Mindy is a sculptor. She has a cylinder of stone with a radius of 3 meters
(m) and a height of
2m. She needs to carve out a sphere of radius
1m from the cylinder. Mindy must cut away
(v)/(3)pi cubic meters
(m^(3)) of stone from the cylinder in order to be left with the sphere. What is the value of
v ?

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

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Scale drawings: word problems

Full solution

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

3

meters

(\mathrm{m})

and a height of

2 \mathrm{~m}

. She needs to carve out a sphere of radius

1 \mathrm{~m}

from the cylinder. Mindy must cut away

\frac{v}{3} \pi

cubic meters

\left(\mathrm{m}^{3}\right)

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

v

?

Given information of Sphere:The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere. $\newline$ Mindy's sphere has a radius of $1$ m, so we substitute $r = 1$ m into the formula.
Substitute Radius into Sphere Formula: Now we calculate the volume of the sphere using the radius. $\newline$ $V = \frac{4}{3}\pi(1)^3 = \frac{4}{3}\pi$ cubic meters.
Given information of Cylinder:The formula for the volume of a cylinder is $V = \pi r^2h$ , where $r$ is the radius and $h$ is the height of the cylinder. $\newline$ Mindy's cylinder has a radius of $3$ m and a height of $2$ m, so we substitute $r = 3$ m and $h = 2$ m into the formula.
Substitute Radius and Height into Cylinder Formula: Now we calculate the volume of the cylinder. $\newline$ $V = \pi (3)^2 \times 2$ $\newline$ $= \pi \times 9 \times 2$ $\newline$ $= \pi \times 18$ $\newline$ $= 18\pi$ cubic meters.
Subtraction of two Volumes: $18\pi - \frac{4}{3}\pi$ $\newline$ $= \frac{54 - 4}{3}\pi$ $\newline$ $= \frac{50}{3}\pi$ $\newline$ The value of $v$ in the expression $\frac{v}{3}\pi$ cubic meters is the coefficient that, when multiplied by $\frac{1}{3}\pi$ , will give us the volume of the sphere. $\newline$ Since we have found the difference of volumes to be $\frac{50}{3}\pi$ cubic meters, we can set up the equation: $\newline$ $\frac{v}{3}\pi = \frac{50}{3}\pi$
Solve for the value of $v$ : To find the value of $v$ , we multiply both sides of the equation by $\frac{3}{\pi}$ to isolate $v$ . $\newline$ $v = \left(\frac{50}{3}\right)\pi \times \left(\frac{3}{\pi}\right) = 50$ $\newline$ So, the value of $v$ is $50$ .

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