Volume of a Cone - Pre-Algebra

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Question

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Answer

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

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

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

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

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

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

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Question

Which of the following is the volume of the above cone?

Answer

The volume of a cone whose height is and whose base has radius is defined by the formula

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

Set :

$V = \frac{1}{3} \pi \cdot 10 ^{2} \cdot 25$

$V = \frac{1}{3} \pi \cdot 100 \cdot 25$

$V = \frac{1}{3} \pi \cdot 2,500$

$V = \frac{2,500 \pi }{3}$ cubic centimeters.

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Question

The standard waffle cone at Cream Canyon Ice Cream Parlor has a diameter of 4 inches. If the height of the cone is 1.5 times the diameter, what is the volume of the cone?

Answer

We must first recall the formula for the volume of a cone.

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

where is the radius and is the height. The problem is that we are not provided with either the height or the radius. However, we are told that the height is 1.5 times the diameter. Since the diameter is 4 inches, we can calculate $1.5\cdot4=6$ . Thus the height of the cone is 6 inches. We must also recall that the radius of a circle (such as the top of an ice cream cone) is simply half the diameter. Therefore, if the diameter is 4 inches, the radius is 2 inches. We now have all of the essential ingredients for volume.

$V=\frac{1}{3}\pi (2)^{2}(6)= \frac{1}{3}\pi (24)=8\pi$

Since our measurements were all in inches, our volume will be in cubic inches. Therefore, the volume of our ice cream cone is $8\pi in.^3$ . Now all we need is a scoop or two of our favorite flavor.

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Question

What is the volume of a cone with a radius of two and a height of three?

Answer

Write the formula to find the volume of the cone.

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

Substitute the radius and height.

$V=\frac{1}{3}\pi (2)^2 (3) = 4\pi$

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Question

What is the volume of a cone with a radius of 5 and a height of 6?

Answer

Write the formula to find the volume of a cone.

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

Substitute the dimensions and solve.

$V=\frac{1}{3} \pi (5)^2 (6) = \pi (5)^2 (2) = 50 \pi$

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Question

Find the volume of a cone with a base area of and a height of .

Answer

The base of a cone has a circular cross section. Given the base area, there is no need to determine the radius.

Write the formula for the volume of a cone.

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

The $\pi r^2$ term represents the base area of the circle.

Substitute all the given values into the volume formula.

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

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Question

What is the volume of a cone with diameter of 2 and a height of 10?

Answer

Write the formula for the volume of a cone.

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

The radius is half the diameter.

Substitute the radius and the height.

$V=\frac{1}{3} \pi (1)^2(10) = \frac{10}{3}\pi$

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Question

Find the volume of a cone with a radius of and a height of .

Answer

Write the volume formula for a cone.

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

Substitute the dimensions.

$V=\frac{1}{3} \pi (10)^2 (1) =\frac{100}{3}\pi$

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Question

Find the volume of the cone that has a radius of $\textup{3 feet}$ and a height of $\textup{2 feet}$ .

Answer

Write the formula to find the volume of a cone.

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

$V=\frac{1}{3} \pi (3 \textup{ feet})^2 (2\textup{ feet}) =6\pi\textup{ feet}^3$

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Question

Find the volume of a cone with a radius of and a height of .

Answer

Write the formula to find the volume of the vone.

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

Substitute the dimensions into the formula.

$V_{cone}= \frac{1}{3} \pi (3)^2(13)$

Expand the terms.

$V_{cone}= \frac{1}{3} \pi (3\cdot3)(13) = \pi (3)(13)$

Multiply the integers for the volume.

$V_{cone} = 39\pi$

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Question

Solve for the volume of a cone with a circular area of $2\pi$ and a height of .

Answer

Write the formula for the volume of a cone.

$V_\textup{cone} = \frac{1}{3} \pi r^2h$

The area of the circular base is represented as $\pi r^2$ in the equation.

Substitute $2\pi$ in replacement of this term. Substitute the height into the volume formula as well.

$V_\textup{cone} = \frac{1}{3} (\pi r^2)h = \frac{1}{3} (2\pi)(6)$

Multiply out each term and leave $\pi$ as is.

$V= \frac{1}{3} (2\pi)(6) = (2\pi)(2) = 4\pi$

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Question

You have an ice cream cone. You want to fill the cone completely with ice cream. What is the volume of ice cream you can fill it with if the height of the cone is 12cm and the diameter is 8cm?

Answer

The formula to find the volume of a cone is

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

where r is the radius and h is the height. We know the diameter of the ice cream cone is 8cm. We also know the radius is half the diameter, so the radius is 4cm. We know the height is 12cm. Using substitution, we get

$V = \pi \cdot (4\text{cm})^2 \cdot \frac{12\text{cm}}{3}$

$V = \pi \cdot 16\text{cm}^2 \cdot 4\text{cm}$

$V = \pi \cdot 64\text{cm}^3$

$V = 64\pi \ \text{cm}^3$

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