Know and Use the Formulas for the Volumes of Cones, Cylinders, and Spheres: CCSS.Math.Content.8.G.C.9

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8th Grade Math › Know and Use the Formulas for the Volumes of Cones, Cylinders, and Spheres: CCSS.Math.Content.8.G.C.9

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A cone has a diameter of and a height of . In cubic meters, what is the volume of this cone?

$48\pi:m^3$

CORRECT

$144\pi:m^3$

$36\pi:m^3$

$54\pi:m^3$

Explanation

First, divide the diameter in half to find the radius.

$\frac{d}{2}=r$

$\frac{12}{2}=6:m$

Now, use the formula to find the volume of the cone.

$\text{Volume}=\frac{1}{3}\pi r^2 h$

$\text{Volume}=\frac{1}{3}\pi \times 6^2\times4=\frac{1}{3}\pi \times 36\times4=\frac{144}{3}\pi=48\pi:m^3$

Calculate the volume of the cylinder provided. Round the answer to the nearest hundredth.

$339.29\textup{ in}^3$

CORRECT

$335.34\textup{ in}^3$

$325.18\textup{ in}^3$

$323.10\textup{ in}^3$

$320\textup{ in}^3$

Explanation

In order to solve this problem, we need to recall the formula used to calculate the volume of a cylinder:

$V=\pi r^2h$

Now that we have this formula, we can substitute in the given values and solve:

$V=\pi3^2(12)$

$V=\pi9(12)$

$V=\pi108$

$V=339.29\textup{ cm}^3$

The height of a cylinder is 3 inches and the radius of the circular end of the cylinder is 3 inches. Give the volume and surface area of the cylinder.

CORRECT

Explanation

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height or:

$Volume=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

$Volume=\pi r^2h=\pi \times 3^2\times 3=84.82in^3$

The surface area of the cylinder is given by:

$A=2\pi r^2+2\pi rh$

where is the surface area of the cylinder, is the radius of the cylinder and is the height of the cylinder. So we can write:

$A=2\pi r^2+2\pi rh$

$A=2\pi (3)^2+2\pi \times 3\times 3$

$A=18\pi+18\pi$

$A=36\pi$

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

$4,500,000\pi \textrm{ cm}^{3}$

CORRECT

$1,125,000\pi \textrm{ cm}^{3}$

$3,375,000\pi \textrm{ cm}^{3}$

$27,000,000\pi \textrm{ cm}^{3}$

$36,000,000\pi \textrm{ cm}^{3}$

Explanation

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute in the volume formula:

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

$V= \frac{4}{3} \cdot \pi \cdot 150^{3}$

$V= \frac{4}{3} \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

Calculate the volume of the sphere provided. Round the answer to the nearest hundredth.

$33.51\textup{ in}^3$

CORRECT

$34.21\textup{ in}^3$

$35\textup{ in}^3$

$36.22\textup{ in}^3$

$38\textup{ in}^3$

Explanation

In order to solve this problem, we need to recall the formula used to calculate the volume of a sphere:

$V=\left ( \frac{4}{3} \right )\pi r^3$

Now that we have this formula, we can substitute in the given values and solve:

$V=\left ( \frac{4}{3} \right )\pi2^3$

$V=\left ( \frac{4}{3} \right )\pi8$

$V=\pi10.66$

$V=33.51\textup{ in}^3$

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

$\frac{1}{18} \pi \textrm{ ft}^{3}$

CORRECT

$\frac{1}{6} \pi \textrm{ ft}^{3}$

$\frac{1}{3} \pi \textrm{ ft}^{3}$

$\frac{1}{12} \pi \textrm{ ft}^{3}$

$\frac{1}{4} \pi \textrm{ ft}^{3}$

Explanation

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = \frac{18}{12} = \frac{3}{2}$ feet

Radius: 4 inches = $4 \div 12 = \frac{4}{12} = \frac{1}{3}$

The volume of a cone is given by the formula

$V = \frac{1}{3} \pi r^{2} h$

Substitute $r = \frac{1}{3}, h = \frac{3}{2}$ :

$V = \frac{1}{3}\cdot \left ( \frac{1}{3} \right ) ^{2}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{3}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{1}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{2} \cdot \pi$

$V = \frac{1}{18} \pi$

A right cone has a volume of $8\pi$ , a height of and a radius of the circular base of . Find .

$\sqrt[3]{2}$

CORRECT

$\sqrt[3]{3}$

$\sqrt{2}$

$2\sqrt[3]{2}$

Explanation

The volume of a cone is given by:

$Volume=\frac{1}{3}\pi r^2h$

where is the radius of the circular base, and is the height; the perpendicular distance from the base to the vertex. Substitute the known values in the formula:

$Volume=\frac{1}{3}\pi r^2h\Rightarrow 8\pi=\frac{1}{3}\pi\times (2t)^2\times 3t$

$\Rightarrow 8\pi=4\pi t^3\Rightarrow t^3=2\Rightarrow t=\sqrt[3]{2}$

The height of a cylinder is two times the length of the radius of the circular end of a cylinder. If the volume of the cylinder is $16\pi$ , what is the height of the cylinder?