Non-Cubic Prisms - Math

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is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

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The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^${2}+9t^{2}$ $+16t^{2}$)$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( $48t^{2}$ +18t ^{2}+24 $t^{2}$)$

$S = 2( 90 $t^{2}$)$

$S = 180 $t^{2}$$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

The length of a box is 3 times the width. Which of the following gives the length (L inches) in terms of the width (W inches) of the box?

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When reading word problems, there are certain clues that help interpret what is going on. The word “is” generally means “=” and the word “times” means it will be multiplied by something. Therefore, “the length of a box is 3 times the width” gives you the answer: L = 3 x W, or L = 3W.

The width of a box, in inches, is 5 inches less than three times its length. Which of the following equations gives the width, W inches, in terms of the length, L inches, of the box?

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We notice the width is “5 inches less than three times its width,” so we express W as being three times its width (3L) and 5 inches less than that is 3L minus 5. In this case, W is the dependent and L is the independent variable.

W = 3L - 5

Find the surface area of the rectangular prism:

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The surface area of a rectangular prism is

Substituting in the given information for this particular rectangular prism.

$\SA=2(13\cdot 9)+2(13\cdot 4)+2(9\cdot 4) \SA=410\ $$\text{units}$^2$$

A rectangular box has two sides with the following lengths:

and

If it possesses a volume of $84$ cm$^{3}$ , what is the area of its largest side?

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The volume of a rectangular prism is found using the following formula:

$V=l\times w\times h$

If we substitute our known values, then we can solve for the missing side.

$84=3\times 4\times x$

Divide both sides of the equation by 12.

$\frac{84}{12}$=\frac{12x}{12}$

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.

$A=l\times w$

$A=4\times 7$

A prism with a square base has a height of feet.

If the edge of the base is feet, what is the volume of the prism?

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The volume of a prism is given as

where

B = Area of the base

and

h = height of the prism.

Because the base is a square, we have

So plugging in the value of B that we found and h that was given in the problem we get the volume to be the following.

$V = 4\cdot4 = $16,ft^3$$

Find the volume of a rectangular prism with a width of , height of and length of .

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The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

Because all the necessary information has been provided to solve for the volume, all that needs to be done is substituting in the values for the variables.

Therefore:

$V = 2:cm \cdot 3:cm \cdot 8:cm$

$V = $6:cm^2$ \cdot 8:cm$

A rectangular prism has the dimensions of , , and . What is the volume of the prism?

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The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

The given information does not explicitly state which side each dimension measurement correlates to on the prism. Volume simply requires the multiplication of the dimensions together.

Volume can be solved for in the following way:

$V = 2:in \cdot 5:in \cdot 7:in$

$V = $10:in^2$ \cdot 7:in$

A small rectangular fish tank has sides that are wide, long, and high. Which formula would not work to find the correct volume of the fish tank?

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In this question the formula that uses addition will not yield the correct volume of the fish tank:

This is the correct answer because in order to find the volume of any rectangular prism one needs to multiply the prism's length, width, and height together.

The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

To restate, is the correct answer because it will NOT yield the correct volume, you would need to multiply by the product of and , not add.

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(12)(15)=90$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=90 \times 20 =1800$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(3)(9)=13.5$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=13.5\times 20 =270$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(5)(9)=22.5$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=22.5\times 14=315$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(6)(12)=36$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=36\times 19=684$

Find the volume of a prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(5)(12)=30$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=30\times 17=510$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(13)(17)=110.5$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=110.5\times 20 =2210$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(13)(19)=123.5$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=123.5\times 28 =3458$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(1)(4)=2$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=2\times 4=8$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(3)(8)=12$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=12\times 10=120$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(9)(10)=45$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=45\times 14=630$

Find the volume of the prism.

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Recall how to find the volume of a prism:

$\text{Volume of Prism}$=$\text{Area of base}$\times$\text{height of prism}$

Find the area of the base, which is a right triangle.

$$\text{Area of Right Triangle}$=\frac{1}{2}$($\text{base}$\times$\text{height}$)$

$$\text{Area of Right Triangle Base}$=\frac{1}{2}$(7)(10)=35$

Now, find the volume of the prism.

$$\text{Volume of Prism}$=35\times 12=420$