GED Math - Volume of a Rectangular Solid

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

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

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

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:

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

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

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

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?

$568.8 \textup{ kg}$

$56.8 8 \textup{ kg}$

$114\textup{ kg}$

$11.4 \textup{ kg}$

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.

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?

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.

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

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?

$182,000 \textrm{ cm}^{3}$

$407,000\textrm{ cm}^{2}$

$429,500\textrm{ cm}^{2}$

$702,000\textrm{ cm}^{2}$

Explanation

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.

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

$28\textup{ in}^3$

$11\textup{ in}^3$

$28\textup{ in}$

$12\textup{ in}^3$

$29\textup{ in}^3$

Explanation

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: $28\textup{ in}^3$

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

$72\textup{ in}^3$

$72\textup{ in}^2$

$72\textup{ in}$

$27\textup{ in}^3$

$36\textup{ in}^3$

Explanation

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: $72\textup{ in}^3$

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?