Non-Cubic Prisms - Math

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Question

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 .

Answer

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.

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Question

A large crate in the shape of a rectangular prism has dimensions 5 feet by 4 feet by 12 feet. Give its volume in cubic yards.

Answer

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

$V = $\frac{5}{3}$ \cdot $\frac{4}{3}$ \cdot $\frac{12}{3}$$

$V = $\frac{5}{3}$ \cdot $\frac{4}{3}$ \cdot $\frac{4}{1}$$

$V = $\frac{80}{9 }$ = 8 $\frac{8}{9 }$ \textrm{ $yd}^{3}$$

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Question

Which is the greater quantity?

(A) The volume of a rectangular solid ten inches by twenty inches by fifteen inches

(B) The volume of a cube with sidelength sixteen inches

Answer

The volume of a rectangular solid ten inches by twenty inches by fifteen inches is

$V = 10 \cdot 20 \cdot15 = 3,000$ cubic inches.

The volume of a cube with sidelength 13 inches is

$V = $16^{3}$ = 16 \cdot 16 \cdot 16 = 4,096$ cubic inches.

This makes (B) greater

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Question

The height of a retangular prism is three times its width; its length is twice its width. Its surface area is 1,078 square inches. Give the width of the box.

Answer

Since the length of the box is twice the width and the height is three times the width, and .

The surface area of the box is

$A=4 $W^{2}$$+12W^{2}$$+6W^{2}$$

To find the width:

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Question

Find the missing edge of the prism when its volume is .

Answer

The goal is to find the height of the rectangular prism with the given information of its width and length. The volume of a rectangular prism is $V = w \cdot l \cdot h$ , where is width and is height.

Because we're given the final volume and two of the three variables, we can substitute in the information we know and solve for the missing variable.

$600= 10 \cdot 12 \cdot h$

$600= 120 \cdot h$

$$\frac{600}{120}$=h$

Therefore, the height of the prism is .

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Question

Find the surface area of the rectangular prism:

Answer

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\ $$\text{units}$^2$$

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Question

The volume of a prism is .

Given the length is and the height is , find the width of the prism.

Answer

The volume of a prism is length times width times height.

We are given the length is 7m and the width is 5m.

So, we plug these into our formula:

$7\cdot5\cdot height=210$ .

We then solve for height:

$height=\frac{210}{7\cdot5}$=\frac{210}{35}$=6$ .

The height is therefore 6m.

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Question

A rectangular prism has a volume of cubic meters. Its length is twice its width, and its height is twice its length. What is the length of the prism?

Answer

If we let $\small w$ be the width of our prism, then since the length is twice the width, our length would be $\small 2w$ . Since the height is twice the length, our height would be $\small 4w$ .

Since the volume of a rectangular prism is simply the product of width, length, and height, we get

$\small $V=lwh=w(2w)(4w)=8w^3$=216$

We then simply solve for the width.

$\small $8w^3$=216$

$\small $w^3$=27$

$\small w=3$

Therefore, our width is 3. Since our length is twice the width, the length is 6.

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Question

A right, rectangular prism has a volume of cubic inches. Its width is inches and its height is inches. What is its length?

Answer

The formula for the volume of a right, rectangular prism is , so substitute the known values and solve for lenght.

$240 = l(5)(6) \rightarrow 240 = 30l \rightarrow8 = l$ . So the length is 8 inches.

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Question

The surface area of a right rectangular prism is square cm. Its length is cm and its height is cm. Find the width of the prism.

Answer

Use the formula for finding the surface area of a right, rectangular prism,

and substitute the known values, and then solve for w.

So,

$358 = 2(12)(w) + 2(w)(5) + 2(12)(5) \rightarrow 358 = 24w + 10w + 120$

$\rightarrow 238 = 34w \rightarrow w = 7$

So, the width is 7 cm.

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Question

Alice is wrapping a rectangular box that measures $5\textup {in}\times 6\textup {in}\times 12\textup {in}$ . How many square feet of wrapping paper does she need?

Answer

The surface area of a rectangular prism is given by

$\textup {SA=2lw+2lh+2wh}$ where is the length, is the width, and is the height.

Let $\textup l=5\textup { in}$ , $\textup {w}=6\textup{ in}$ , and $\textup {h}=12\textup { in}$

So the equation to solve becomes $\textup SA=2\cdot 5\cdot 6+2\cdot 5\cdot 12+2\cdot 6\cdot 12$ or $324\textup{ $in}^{2}$$

However the question asks for an answer in square feet. Knowing that $144\textup { $in}^{2}$=1\textup { $ft}^{2}$$ we can convert square inches to square feet. It will take $2.25\textup{ $ft}^{2}$$ of paper to wrap the present.

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Question

Find the surface area of the rectangular prism:

Answer

To find the surface area of a prism, the problem can be approached in one of two ways.

1. Through an equation that uses lateral area
2. Through finding the area of each side and taking the sum of all the faces

Using the second method, it's helpful to realize rectangular prisms contain faces. With that, it's helpful to understand that there are pairs of sides. That is, there are two faces with the same dimensions. Therefore, we really only have three sides for which we need to calculate areas:

Faces 1 & 2:

$15\cdot8=120$

Faces 3 & 4:

$6\cdot8=48$

Faces 5 & 6:

$15\cdot6=90$

Now, we can add up the areas of all six sides:

The surface area is .

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Question

Find the surface area of the rectangular prism:

Answer

Surface area means the entire area that all the sides of a prism take up.

The surface area can be calculated in one of two ways. One way involves using an equation for lateral area. The other method involves taking the area of all the sides and summing the areas.

Using the latter of the two methods:

It's helpful to understand that rectangular prisms have three pairs of sides with the same dimensions, making up the total of six faces. This means that only three novel calculations for individual areas of faces need to be calculated.

Faces 1 & 2:
$4 \cdot 7 = 28$
$4 \cdot 7 = 28$

Faces 3 & 4:
$7 \cdot 18 = 126$
$7 \cdot 18 = 126$

Faces 5 & 6:
$4 \cdot 18 = 72$
$4 \cdot 18 = 72$

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Question

The height of a ramp is meters and spans an meter walkway. Sara wants to paint the ramp to match her house, but needs to know the surface area.

What is the surface area of the ramp to the nearest meter?

Answer

Since the ramp forms a 45-45-90 triangle, the base of the ramp is equal to the height. So the area of the triangle is $$\frac{3\cdot 3}{2}$=4.5$ meters. The area of the two triangles would be $4.5\cdot2=9$ meters. The other sides of the ramp are rectangles. Two of the rectangles are the same with one having a different length due to the hypotenuse of the triangle. The two that are the same have a length of 3 and a width of 8. The area for each of these is $3\cdot8=24$ meters. Since there are 2 of these, we mulitply 24 by 2. For the last triangle, we must find the hypotenuse of the triangle. Since it is a 45-45-90, the hypotenuse is the base multiplied by $\sqrt{2}$ . Therefore the last rectangle's is $3$\sqrt{2}$\cdot8=33.9$ meters. The find the surface area, all of the areas must be added together. Triangles+rectangles= meters. To the nearest whole meter, the answer is 91 meters.

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Question

Kate has an open top box that has the following dimensions: inches tall, inches wide, and inches long.

In square inches, how much wrapping paper would it take to cover the box?

Answer

Since the box has an open top, the surface area is calculated by finding the four sides plus the floor of the box.

The short sides' areas are calculated by multiplying height times width times two, for the two sides: $6\cdot7\cdot2=84$ .

The longer sides' areas are calculated by multiplying height times length times two, for the two sides: $6\cdot12\cdot2=144$ .

Now, we calculate the area of the floor of the box by multiplying length times width times one, for the only floor and no top: $7\cdot12\cdot1=84$ .

Lastly, we add the areas together to calculate the total surface area of the open-top box: ,

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Question

A small rectangular jewelry box has two square ends with areas of 36 square centimeters, and a width of 10 centimeters. What is the surface area of the outside of the jewelry box.

Answer

To find the surface area of the rectangular box we just need to add up the areas of all six sides. We know that two of the sides are 36 square centimeters, that means we need to find the areas of the four mising sides. To find the area of the missing sides we can just multiply the side of one of the squares (6 cm) by the width of the box:

$6 \times 10 = 60 ; $cm^2$$

But remember we have four of these rectangular sides:

$4 \times 60 = 240 ; $cm^2$$

Now we just add the two square sides and four rectangular sides to find the total surface area of the jewelry box:

That is the total surface area!

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Question

Find the surface area of the regular hexagonal prism.

Answer

In order to find the surface area, you will need to add up the areas of each face of the prism.

This hexagonal prism has two regular hexagons as its bases.

Recall how to find the area of a regular hexagon:

$$\text{Area of Regular $Hexagon}$=\frac{3\sqrt3}{2}$($\text{side}$^2$)$

Now, since we have two hexagons, we can multiply the area by to get the area of both bases.

$\text{Area of $Bases}$=3\sqrt3($\text{side}$)^2$

Next, this prism has rectangles that make up its sides.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}$=$\text{length}$\times$\text{width}$

In this prism, the length is also the side of the hexagon, and the width of the rectangle is the height of the prism.

Now, multiply the area of the rectangle by to find the total area of all of the sides of the prism.

$$\text{Area of Prism Sides}$=6($\text{side}$)($\text{height}$)$

Add together the area of the sides and of the bases to find the surface area of the prism.

$\text{Surface Area of Prism}$=$\text{Area of Bases}$+$\text{Area of Sides}$

$$\text{Surface Area of $Prism}$=3\sqrt3($\text{side}$)^2$+6($\text{side}$)($\text{height}$)$

Plug in the given values to find the surface area of the prism.

$$\text{Surface Area of $Prism}$=3\sqrt3(18)^2$+6(18)(24)=4275.55$

Make sure to round to places after the decimal.

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Question

Find the surface area of the regular hexagonal prism.

Answer

In order to find the surface area, you will need to add up the areas of each face of the prism.

This hexagonal prism has two regular hexagons as its bases.

Recall how to find the area of a regular hexagon:

$$\text{Area of Regular $Hexagon}$=\frac{3\sqrt3}{2}$($\text{side}$^2$)$

Now, since we have two hexagons, we can multiply the area by to get the area of both bases.

$\text{Area of $Bases}$=3\sqrt3($\text{side}$)^2$

Next, this prism has rectangles that make up its sides.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}$=$\text{length}$\times$\text{width}$

In this prism, the length is also the side of the hexagon, and the width of the rectangle is the height of the prism.

Now, multiply the area of the rectangle by to find the total area of all of the sides of the prism.

$$\text{Area of Prism Sides}$=6($\text{side}$)($\text{height}$)$

Add together the area of the sides and of the bases to find the surface area of the prism.

$\text{Surface Area of Prism}$=$\text{Area of Bases}$+$\text{Area of Sides}$

$$\text{Surface Area of $Prism}$=3\sqrt3($\text{side}$)^2$+6($\text{side}$)($\text{height}$)$

Plug in the given values to find the surface area of the prism.

$$\text{Surface Area of $Prism}$=3\sqrt3(10)^2$+6(10)(12)=1239.62$

Make sure to round to places after the decimal.

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Question

Find the surface area of the regular hexagonal prism.

Answer

In order to find the surface area, you will need to add up the areas of each face of the prism.

This hexagonal prism has two regular hexagons as its bases.

Recall how to find the area of a regular hexagon:

$$\text{Area of Regular $Hexagon}$=\frac{3\sqrt3}{2}$($\text{side}$^2$)$

Now, since we have two hexagons, we can multiply the area by to get the area of both bases.

$\text{Area of $Bases}$=3\sqrt3($\text{side}$)^2$

Next, this prism has rectangles that make up its sides.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}$=$\text{length}$\times$\text{width}$

In this prism, the length is also the side of the hexagon, and the width of the rectangle is the height of the prism.

Now, multiply the area of the rectangle by to find the total area of all of the sides of the prism.

$$\text{Area of Prism Sides}$=6($\text{side}$)($\text{height}$)$

Add together the area of the sides and of the bases to find the surface area of the prism.

$\text{Surface Area of Prism}$=$\text{Area of Bases}$+$\text{Area of Sides}$

$$\text{Surface Area of $Prism}$=3\sqrt3($\text{side}$)^2$+6($\text{side}$)($\text{height}$)$

Plug in the given values to find the surface area of the prism.

$$\text{Surface Area of $Prism}$=3\sqrt3(8)^2$+6(8)(12)=908.55$

Make sure to round to places after the decimal.

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Question

Find the surface area of the regular hexagonal prism.

Answer

In order to find the surface area, you will need to add up the areas of each face of the prism.

This hexagonal prism has two regular hexagons as its bases.

Recall how to find the area of a regular hexagon:

$$\text{Area of Regular $Hexagon}$=\frac{3\sqrt3}{2}$($\text{side}$^2$)$

Now, since we have two hexagons, we can multiply the area by to get the area of both bases.

$\text{Area of $Bases}$=3\sqrt3($\text{side}$)^2$

Next, this prism has rectangles that make up its sides.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}$=$\text{length}$\times$\text{width}$

In this prism, the length is also the side of the hexagon, and the width of the rectangle is the height of the prism.

Now, multiply the area of the rectangle by to find the total area of all of the sides of the prism.

$$\text{Area of Prism Sides}$=6($\text{side}$)($\text{height}$)$

Add together the area of the sides and of the bases to find the surface area of the prism.

$\text{Surface Area of Prism}$=$\text{Area of Bases}$+$\text{Area of Sides}$

$$\text{Surface Area of $Prism}$=3\sqrt3($\text{side}$)^2$+6($\text{side}$)($\text{height}$)$

Plug in the given values to find the surface area of the prism.

$$\text{Surface Area of $Prism}$=3\sqrt3(7)^2$+6(7)(13)=800.61$

Make sure to round to places after the decimal.

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