Volume of a Rectangular Solid - GED Math

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Question

An aquarium takes the shape of a rectangular prism 60 centimeters high, 60 centimeters wide, and 120 centimeters long. One-fourth of a cubic meter of water is poured into the aquarium after it has been emptied. How much more water can it hold?

Answer

One cubic meter of water is equal to $100^{3} = 1,000,000$ cubic centimeters; one-fourth of a cubic meter is 250,000 centimeters. The volume of the aquarium is

$60 \times 60 \times 120 = 432,000$ cubic centimeters.

Therefore, after having one-fourth of a cubic meter of water poured in, there is room left for

cubic centimeters of water.

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Question

A large aquarium has a rectangular base nine meters square and is ten and one-half meters high. The eight inlet pipes used to fill the aquarium does so at a rate of 200 liters per minute each. To the nearest hour, how long does it take for all eight pipes working together to fill the aquarium to 80% capacity?

You will need the conversion factor 1 liter = 1,000 cubic centimeters.

Answer

The aquarium is a rectangular prism with dimensions 9 meters by 9 meters by 10.5 meters, each of which can be converted to centimeters by multiplying by 100. So the dimensions are 900 cm x 900 cm x 1,050 cm, and the volume is the product of the three, or

$900 \times 900 \times 1,050 = 850,500,000 \textup{ cm}^{3}$

or

$850,500,000 \textup{ cm}^{3} \div 1,000 = 850,500 \textup{ L}$

80% of this is

$850,500 \textup{ L } \times 80 % = 680,400\textup{ L}$

Each pipe fills the aquarium at 200 L per minute, so the eight pipes working together fill it at a rate of 1,600 L per minute. Divide, and the aquarium is filled at 80% capacity in

$680,400\textup{ L } \div 1,600 \textup{ L \ min}= 425.25\textup{ min}$

In hours, this is

$425.25 \div 60 \approx 7.1$

so 7 hours is the correct response.

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Question

If a swimming pool is rectangular, and its base area is 20 feet squared, what is the volume if the height of the pool is 8 feet?

Answer

The volume of the rectangular solid can be written as:

The length times width constitutes the base of the rectangular solid, and is given in the question.

Substitute the known dimensions into the formula.

$V= 20\textup{ ft}^2(8 \textup{ ft}) = 160 \textup{ ft}^3$

The answer is: $160 \textup{ ft}^3$

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Question

A rectangular wood block has the dimensions of 6 inches, 1 foot, and 8 inches. What is the volume in feet?

Answer

Convert all the dimensions into feet.

$6 \textup{ inches} = \frac{1}{2} \textup{ ft}$

$8 \textup{ inches} = \frac{8}{12} \textup{ ft}=\frac{2}{3} \textup{ ft}$

Write the formula for the volume of the block.

Substitute the dimensions and solve for volume.

$V= (\frac{1}{2} \textup{ ft})(\frac{2}{3} \textup{ ft})(1\textup{ ft}) = \frac{1}{3}\textup{ ft}^3$

The answer is: $\frac{1}{3}\textup{ ft}^3$

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Question

Find the volume of a rectangular wood block with a length of , width of $\frac{1}{2}$ , and a height of $\frac{1}{3}$ .

Answer

Write the formula for the volume of a rectangular prism.

Substitute the dimensions into the formula.

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

The answer is:

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Question

Determine the volume of a brick in inches if its dimensions are 3 inches by 6 inches by 2 inches.

Answer

The volume of a rectangular prism can be represented as:

Substitute the dimensions into the formula.

$V = (3 \textup{ in}) (6 \textup{ in}) (2 \textup{ in}) = 36 \textup{ in}^3$

Be sure that inches is cubed for volume.

The answer is: $36 \textup{ in}^3$

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Question

What is the volume of a brick with a length of 6 inches, width of 4 inches, and a height of 3 inches?

Answer

The volume of a brick is similar to finding the volume of a rectangular solid.

The volume for a rectangular solid is:

Substitute the dimensions.

$A =(6\textup{ in})(4\textup{ in})(3\textup{ in}) = 72\textup{ in}^3$

The answer is: $72\textup{ in}^3$

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Question

Find the volume of the rectangular solid with a length, width, and height of 4,5,and 6, respectively.

Answer

Write the volume for the rectangular solid.

Substitute the dimensions into the formula.

$V = 4\times 5\times 6 = 120$

The answer is:

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Question

Find the volume of a rectangular solid if the length, width, and height are 6, 12, and 20, respectively.

Answer

Write the volume for a rectangular solid.

Substitute the dimensions into the formula.

$V= 6\times 12\times 20 = 1440$

The answer is:

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Question

What is the volume of a book with a length of 7 inches, width of 4 inches, and a height of 1 inches?

Answer

The book resembles a rectangular solid. Write the formula for the volume of a rectangular solid.

Substitute the dimensions into the equation.

$V= (7 \textup{ in})(4\textup{ in})(1\textup{ in}) = 28\textup{ in}^3$

The answer is: $28\textup{ in}^3$

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Question

Tammy has an aquarium in the shape of a rectangular prism. The aquarium has the following dimensions: $12 in \times 16 in \times 30 in$ . In order for her to properly clean the aquarium, she must remove two-thirds of the water in the aquarium. In cubic inches, how much water must she remove?

Answer

Start by finding the volume of the rectangular prism.

$\text{Volume}=\text{length}\times \text{width}\times \text{height}$

For the given dimensions,

$\text{Volume}=12\times 16 \times 30=5760$

Since Tammy needs to remove two-thirds of the water, we will need to find two-thirds of the volume.

$(5760)\frac{2}{3}=3840$

Tammy must remove of water.

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Question

If a brick is a rectangular sold, what is its volume if its base area is 4, and the height is 5?

Answer

Write the formula for the area of a rectangular solid.

The base area consists of , which means we can substitute the area as replacement of the two variables.

The answer is:

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Question

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?

Answer

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:

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

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Question

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?

Answer

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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Question

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

Answer

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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Question

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?

Answer

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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Question

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

Answer

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.

So, plug in and multiply to get:

So, our answer is:

Coincidentally the same as our surface area

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Question

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

Answer

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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