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GED Math : 3-Dimensional Geometry

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Example Question #11 : Volume Of A Rectangular Solid

For an art project, Amy needs to paint a rectangular box with the dimensions $4in\times 6in \times 10in$ red, blue, and yellow. Each color must take up one-third of the painted surface. In square inches, how much blue paint is needed?

Possible Answers:

78.33

66.33

82.67

85.67

Correct answer:

82.67

Explanation:

Since Amy is painting the outside of a box, we will need to find the surface area of the box.

Recall how to find the surface area of a rectangular prism:

$\text{Surface Area}=2(wl+hl+hw)$ , where is the width, is the height, and is the length.

Because we are only interested in the amount of blue paint that Amy will be painting, we know that we will need to find one-third of the surface area.

$\text{Area of Blue Paint}=\frac{\text{Surface Area}}{3}=\frac{2(wl+hl+hw)}{3}$

Plug in the dimensions of the box to find the area of the blue paint.

$\text{Area of Blue Paint}=\frac{2((4)(6)+(6)(10)+(4)(10))}{3}=\frac{248}{3}=82.67$

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Example Question #11 : Volume Of A Rectangular Solid

A rectangular prism has as its three dimensions , , and x+ 6 . Give its volume in terms of .

Possible Answers:

$2 x ^{3} +6x ^{2}$

4x+6

4x+12

$2 x ^{3} +12 x ^{2}$

Correct answer:

$2 x ^{3} +12 x ^{2}$

Explanation:

The volume of a rectangular prism is equal to the product of its three dimensions, so here,

$V= x \cdot 2x \cdot (x+6)$

$=2 \cdot x \cdot x \cdot (x+6)$

$=2 x ^{2} (x+6)$

Apply the distribution property, multiplying $2x^{2}$ by each of the expressions in the parentheses:

$V =2 x ^{2} \cdot x+2 x ^{2} \cdot 6$

$=2 x ^{2+1} +2 \cdot 6 \cdot x ^{2}$

$=2 x ^{3} +12 x ^{2}$

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Example Question #15 : Volume Of A Rectangular Solid

You are building a metal crate to hold fishing equipment. If the crate will be 1.5 ft long, 2 feet tall, and 5 feet wide, what will its volume be?

Possible Answers:

V=15ft^3

V=10.5ft^3

V=8.5ft^3

V=115ft^3

Correct answer:

V=15ft^3

Explanation:

You are building a metal crate to hold fishing equipment. If the crate will be 1.5 ft long, 2 feet tall, and 5 feet wide, what will its volume be?

We are asked to find the volume of a rectangular solid. In this case it is a metal crate, but it is essentially a rectangular solid. To find its volume, use the following formula:

V=l*w*h

Where, l, w, and h are the length, width and height.

V=1.5ft*2ft*5ft=1.5ft*10ft^2=15ft^3

V=15ft^3

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Example Question #16 : Volume Of A Rectangular Solid

Prism

One cubic centimeter of pure iron is about $7.9 \textup{ grams}$ in mass.

Using this figure, what is the mass, in kilograms, of the above iron bar?

Possible Answers:

$56.8 8 \textup{ kg}$

$568.8 \textup{ kg}$

$114\textup{ kg}$

$11.4 \textup{ kg}$

Correct answer:

$568.8 \textup{ kg}$

Explanation:

First, convert the dimensions of the prism to centimeters. One meter is equal to 100 centimeters, so multiply by this conversion factor:

$0.8 \textup{ m } \times 100 \textup{ cm /m} = 80 \textup{ cm}$

$0.3 \textup{ m } \times 100 \textup{ cm /m} =30 \textup{ cm}$

The dimensions of the prism are 80 centimeters by 30 centimeters by centimeters; multiply these dimensions to find the volume:

$V = 80 \textup{ cm} \times 30 \textup{ cm} \times 30 \textup{ cm}$

$= 72,000 \textup{ cm}^{3}$

Using the given mass of 7.9 grams per cubic centimeter, multiply:

$M = 72,000 \textup{ cm}^{3} \times 7.9 \textup{ g / cm}^{3} = 568,800 \textup{ g}$

One kilogram is equal to 1,000 grams, so divide by this conversion factor:

$M = 568,800 \textup{ g} \div 1,000 \textup{ g / kg} = 568.8 \textup{ kg}$ ,

the correct mass of the prism.

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Example Question #41 : 3 Dimensional Geometry

Find the volume of a rectangular prism with the following dimensions: 6 ft by 12 ft by 4 ft.

Possible Answers:

288 ft^3

198 ft^3

148 ft^3

208 ft^3

Correct answer:

288 ft^3

Explanation:

Find the volume of a rectangular prism with the following dimensions: 6 ft by 12 ft by 4 ft.

To find the volume of a rectangular prism, simply multiply the length by the width by the height.

V=l*w*h

So, plug in and multiply to get:

V=6ft*4ft*12ft=24ft^2*12ft=288ft^3

So, our answer is:

288 ft^3

Coincidentally the same as our surface area

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Example Question #42 : 3 Dimensional Geometry

What is the volume of a box with length of 3 feet, width of 5 feet, and height of 2 feet?

Possible Answers:

15 feet squared

30 feet squared

10 feet squared

12 feet squared

7 feet squared

Correct answer:

30 feet squared

Explanation:

The equation for the volume of a rectangular prism is

$V=l\cdot w \cdot h$

So we simply input our dimensions

$V=5\cdot2\cdot3=30$

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Example Question #1 : Volume Of A Cylinder

One cubic foot is equal to (about) 7.5 gallons.

A circular swimming pool has diameter 60 feet and depth five feet throughout. Using the above conversion factor, how many gallons of water does it hold?

Use 3.14 for $\pi$ .

Possible Answers:

$105,975\textrm{ gal }$

$211,950\textrm{ gal }$

$7,065 \textrm{ gal }$

$24,320\textrm{ gal}$

Correct answer:

$105,975\textrm{ gal }$

Explanation:

The pool can be seen as a cylinder with depth (or height) 5 feet, and a base with diameter 60 feet - and radius half this, or 30 feet. The capacity of the pool is the volume of this cylinder, which is

$V = \pi r ^{2} h = 3.14 \times 30^{2} \times 5 = 14,130$ cubic feet.

One cubic foot is equal to 7.5 gallons, so multiply:

$14,130 \times 7.5 = 105,975$ gallons

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Example Question #2 : Volume Of A Cylinder

A cylindrical bucket is one foot high and one foot in diameter. It is filled with water, which is then emptied into an empty barrel three feet high and two feet in diameter. What percent of the barrel has been filled?

Possible Answers:

$12 \%$

$16\frac{2}{3} \%$

$18 \%$

$8 \frac{1}{3} \%$

Correct answer:

$8 \frac{1}{3} \%$

Explanation:

The volume of a cylinder is

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

The bucket has height h = 1 and diameter 1, and,subsequently, radius $r = \frac{1}{2}$ ; its volume is

$V = \pi r^{2}h = \pi \cdot \left (\frac{1}{2} \right ) ^{2} \cdot 1 =\frac{1}{4} \pi$ cubic feet

The barrel has height h = 3 and diameter 2,and, subsequently, radius r = 1 ; its volume is

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

The volume of the bucket is

$\frac{\frac{1}{4}\pi}{3 \pi} \times 100 \%$

$=\left ( \frac{1}{4}\pi \div 3 \pi \right )\times 100 \%$

$=\left ( \frac{\pi}{4}\times \frac{1}{3 \pi} \right )\times 100\%$

$= \frac{1}{12} \times 100\% = 8 \frac{1}{3} \%$

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Example Question #1 : Volume Of A Cylinder

A water tank takes the shape of a closed cylinder whose exterior has height 30 feet and a base with radius 10 feet; the tank is three inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

You may use 3.14 for $\pi$ .

Possible Answers:

$9,000 \textrm{ ft}^{3}$

$8,800 \textrm{ ft}^{3}$

$9,400 \textrm{ ft}^{3}$

$8,400 \textrm{ ft}^{3}$

Correct answer:

$8,800 \textrm{ ft}^{3}$

Explanation:

Three inches is equal to 0.25 feet, so the height of the interior of the tank is

$30 - 2 \times 0.25 = 30 - 0.5 = 29.5$ inches.

The radius of the interior of the tank is

10 - 0.25 = 9.75 inches.

The amount of water the tank holds is the volume of the interior of the tank, which is

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

$V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5$

$V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5 \approx 8,806$ cubic feet.

This rounds to 8,800 cubic feet.

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Example Question #1 : Volume Of A Cylinder

Barrel

The above diagram is one of a cylindrical tub. The company wants to make a cylindrical tub with three times the volume, but whose base is only twice the radius. How high should this new tub be?

Possible Answers:

$26 \frac{2}{3}\textrm{ in}$

$53 \frac{1}{3}\textrm{ in}$

$60\textrm{ in}$

$30\textrm{ in}$

Correct answer:

$30\textrm{ in}$

Explanation:

The volume of the given tub can be expressed using the following formula, setting r = 25 and h= 40 :

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

$V = \pi \cdot 25^{2} \cdot 40$

$V = 25,000 \pi$ cubic inches.

The new tub should have three times this volume, or

$V = 25,000 \pi \cdot 3 = 75,000 \pi$ cubic inches.

The radius is to be twice that of the above tub, which will be

$r = 25 \cdot 2 = 50$ inches.

The height can therefore be calculated as follows:

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

$\pi \cdot 50^{2} \cdot h = 75,000 \pi$

$2,500 \pi h = 75,000 \pi$

$h = \frac{75,000 \pi}{2,500 \pi} = 30$ inches

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GED Math : 3-Dimensional Geometry

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #11 : Volume Of A Rectangular Solid

Example Question #11 : Volume Of A Rectangular Solid

Example Question #15 : Volume Of A Rectangular Solid

Example Question #16 : Volume Of A Rectangular Solid

Example Question #41 : 3 Dimensional Geometry

Example Question #42 : 3 Dimensional Geometry

Example Question #1 : Volume Of A Cylinder

Example Question #2 : Volume Of A Cylinder

Example Question #1 : Volume Of A Cylinder

Example Question #1 : Volume Of A Cylinder

All GED Math Resources

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