Geometry - Pre-Algebra

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Question

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Answer

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet = $4 \times 12 = 48$ inches

Sidelength of the base: 3 feet = $3 \times 12 = 36$ inches

The volume of a pyramid is

$V = \frac{1}{3} Bh$

Since the base is a square, we can replace $B = s^{2}$ :

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

Substitute

$V = \frac{1}{3} \cdot 36 ^{2}\cdot 48$

The pyramid has volume 20,736 cubic inches.

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Question

The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.

Answer

Convert each of the measurements from feet to inches by multiplying by .

Height: $2 \times 12 = 24$ inches

Sidelength of base: $3 \times 12 = 36$ inches

The base of the pyramid has area

$B = s^{2} = 36^{2} = 1,296$ square inches.

Substitute into the volume formula:

$V = \frac {1}{3} Bh$

$V = \frac {1}{3} \cdot 1,296 \cdot 24 =10,368$ cubic inches

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Question

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

Answer

The perimeter of the square base, feet, is equivalent to $3 \times 12 = 36$ inches; divide by to get the sidelength of the base - and the height: $36 \div 4 = 9$ inches.

The area of the base is therefore $B = s^{2} = 9^{2} = 81$ square inches.

In the formula for the volume of a pyramid, substitute :

$V = \frac{1}{3} Bh = \frac{1}{3}\cdot 81 \cdot 9 = 243$ cubic inches.

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Question

What is the volume of a pyramid with the following measurements?

$length=7; width=6;height=9;slant\ length=11$

Answer

The volume of a pyramid can be determined using the following equation:

$V=\frac{1}{3}lwh=\frac{1}{3}(7)(6)(9)=126$

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Question

Principal O'Shaughnessy has a paperweight in the shape of a pyramid with a square base. If one side of the base has a length of 4cm and the height of the paperweight is 6cm, what is the volume of the paperweight?

Answer

We begin by recalling the volume of a pyramid.

$V=\frac{1}{3}Ah$

where is the area of the base and is the height.

Since the base is a square, we can find the area by squaring the length of one of the sides.

Given the height is 6cm, we can now calculate the volume.

$V=\frac{1}{3}(16)(6)=32$

Since all of the measurements were in centimeters, our volume will be in cubic centimeters.

Therefore, the volume of Principal O'Shaughnessy's paperweight is .

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Question

The volume of a square pyramid is $162 m^{3}$ . If a side of the square base measures . What is the height of the pyramid?

Answer

The formula for the volume of a pyramid is $\frac{\left ( B \times h\right )}{3}$ , where is the area of the base and is the height.

Using this formula,

= Area of the base, which is nothing but area of square with side . $9m \times 9m = 81 m^{2}$

Now, $\frac{81m^{2} \times h}{3} = 162 m^{3}$ when simplified, you get .

Hence, the height of the pyramid is .

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Question

The pyramid has a length, width, and height of respectively. What is the volume of the pyramid?

Answer

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (L\times W\times H)$

Substitute the dimensions and solve.

$V=\frac{1}{3} (2\times 4\times 6) = \frac{1}{3}(48)=16$

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Question

If the base area of the pyramid is , and the height is , what is the volume of the pyramid?

Answer

Write the volume formula for the pyramid.

$V=\frac{1}{3} (L\times W\times H)$

The base area is represented by $L\times W$ .

Substitute the knowns into the formula.

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

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Question

Find the volume of a pyramid with a length of 4, width of 7, and a height of 3.

Answer

Write the formula to find the area of a pyramid.

$V=\frac{1}{3} (L\times W \times H)$

Substitute the dimensions.

$V=\frac{1}{3} (4\times 7\times 3) = 28$

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Question

Find the volume of a pyramid if the length, base, and height are $\textup{10, 20, and 40,}$ respectively.

Answer

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (LWH)$

Substitute the dimensions and solve for the volume.

$V=\frac{1}{3} (10)(20)(40) = \frac{8000}{3}$

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Question

Find the volume of a pyramid with a length of 2, width of 6, and a height of 9.

Answer

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (\textup{ Length} \times\textup{ Width} \times\textup{ Height})$

Substitute the given length, width, and height.

$V=\frac{1}{3} (2\times6\times9)$

Rewrite the inside the parentheses as a factor of .

$V=\frac{1}{3} (2\times (3 \times 2)\times9)$

Cancel the fraction with the three and multiply the terms to get the volume.

$V=2\times 2\times 9 = 36$

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Question

Find the volume of a pyramid if the dimensions of the length, width, and height are , respectively.

Answer

Write the volume formula for a pyramid.

$V=\frac{1}{3}( \textup{Length} \times \textup{Width } \times \textup{Height})$

Plug in the dimensions.

$V=\frac{1}{3}( 3\times5 \times 10)=\frac{3\times5 \times 10}{3}$

Cancel out the three on the numerator and denominator.

$V=5\times 10$

Multiply.

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Question

Find the volume of a pyramid with a length of 6cm, a width that is half the length, and a height that is two times the length.

Answer

The formula for volume of a pyramid is

$V =\frac{ l \cdot w \cdot h}{3}$

where l is the length, w is the width, and h is the height. We know the length is 6cm. The width is half the length, so the width is 3cm. The height is two times the length, so the height is 12cm. Using this information, we substitute. We get

$V =\frac{ 6\text{cm} \cdot 3\text{cm} \cdot 12\text{cm}}{3}$

$V =\frac{ 216}{3}\text{cm}^3=72 \text{cm}^3$

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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

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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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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