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# Area and Volume of Similar Solids

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Understand Scale Relationships
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Have you ever thought about scale relationships? Take a look at this dilemma.

Tim has a cube with a side length of 4 inches. He has a similar cube with dimensions that are twice the first cube. How does the volume of the larger cube compare to the volume of the smaller cube?

This Concept will show you how to tackle problems like this one.

### Guidance

We can compare the scale relationships of distance, area and volume when looking at three – dimensional figures. If you think back to other math classes, you will remember some of these three – dimensional figures such as a prism or a pyramid. When you compare different measurements, you will see the proportional relationships between them.

Let’s look at a situation involving volume.

Brooke has a scale model of a warehouse. A storage unit is shaped like a rectangular prism and has the dimensions 4 in. by 3 in. by 6 in. If the scale of the model is 0.5 in. = 2 ft, what are the actual dimensions of the storage unit? What is the volume?

First, notice that there are two parts to this problem. The first part is figuring out the actual dimensions given that Brooke has a scale model. The second part is figuring out the volume.

First use a proportion to find the actual dimensions of the storage unit.

Write the scale as the first ratio, and the scale and unknown actual dimension of the storage unit as the second ratio.

$\frac{0.5 \ inch}{2 \ feet} &= \frac{4 \ inches}{x \ feet} ~~ & ~~ \frac{0.5 \ inch}{2 \ feet} & = \frac{3 \ inches}{x \ feet} ~~ & ~~ \frac{0.5 \ inch}{2 \ feet} & = \frac{6 \ inches}{x \ feet}\\(0.5)x &= 4(2) & (0.5)x &= 3(2) & (0.5)x &= 6(2)\\ 0.5x &= 8 & 0.5x &= 6 & 0.5x &= 12\\ x &= 16 & x &= 12 & x &= 24$

The actual dimensions of the storage unit are 16 feet by 12 feet by 24 feet. This is the length, width and height of the storage unit.

Now that you know the actual dimensions, you can find the volume.

$V &= lwh\\V &= (16 \ feet)(12 \ feet)(24 \ feet)\\A &= 4,608 \ feet^3$

The volume of the storage unit is 4,608 cubic feet.

There is a relationship between the area of the base of the prism and the volume of the prism. Let's take a look at how the area of the base of the prism relates to the volume of the prism.

$A &= lw\\A &= 16(12)\\A &= 192 \ sq.feet$

If we write the volume as a ratio with the area of the base, we will find something very interesting.

$\frac{4608}{192}$

Now divide the numerator by the denominator.

The answer is 24 feet. This is the measurement of the height of the prism.

This means that there is a relationship between the area of a three – dimensional figure, its height and its volume. The measurements are related and in proportion to one another.

Use what you have learned to answer each question above.

#### Example A

Find the volume of a prism with a length of 16 feet, a width of 12 feet and a height of 18 feet.

Solution: 3456 cubic feet

#### Example B

Now find the area of the base of the prism.

Solution: 192 feet

#### Example C

Next, write a ratio comparing the volume to the area of the and simplify.

Solution: $\frac{3456}{192}$ = $18$ feet

Now let's go back to the dilemma from the beginning of the Concept.

First, find the dimensions of the larger cube.

The problem states that the dimensions are twice those of the first cube. That means they are scaled up by a factor of 2. So the side length of the larger cube is $4 \ inches \times 2 = 8 \ inches$ .

Now find the volume of both cubes and compare.

Volume of smaller cube:

$V &= lwh\\V &= (4 \ inches)(4 \ inches)(4 \ inches)\\V &= 64 \ inches^3$

Volume of larger cube:

$V &= lwh\\V &= (8 \ inches)(8 \ inches)(8 \ inches)\\V &= 512 \ inches^3$

Next compare the two volumes.

You want to know how the volume of the larger cube compares to the volume of the smaller cube.

Write a ratio comparing the two volumes.

$\frac{512 \ inches^3}{64 \ inches^3}=8$

The volume of the larger cube is 8 times larger than the volume of the smaller cube.

### Vocabulary

Two – Dimensional
A figure drawn in two dimensions is only drawn using length and width.
Three – Dimensional
A figure drawn using length, width and height or depth.
Scale Model
a model that represents a three – dimensional space.

### Guided Practice

Here is one for you to try on your own.

Prove that the height of the following prism can be found by using a ratio of volume to area.

A prism with a length of 6 inches, a width of 5 inches and a height of 9 inches.

Solution

First, let's find the volume of the prism.

$V = lwh$

$V =(6)(5)(9)$

$V = 270$ cubic inches

Now let's find the area of the base.

$A = lw$

$A = (5)(6)$

$A = 30$ sq. inches

Next, we can write a ratio comparing volume to area.

$\frac{270}{30}$

To prove the relationship, we simplify this ratio. When we do this, we should find the height.

$9$ inches

Our work is complete.

### Practice

Directions: Solve each problem.

1. A cube measures 8 feet on each side. A similar cube has dimensions that are twice as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
2. A cube measures 3 inches on each side. A similar cube has dimensions that are half that of the other cube. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison
3. A scale model of a sandbox has dimensions 0.5 inch by 3 inches by 4 inches. If the scale of the model is $1 \ inch = 1 \ foot$ , what is the volume of the actual sandbox?
4. A cube measures 5 inches on each side. A similar cube has dimensions that are 3 times as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
5. A shipping box measures 16 inches by 12 inches by 8 inches. A second box has a similar size but each dimension is $\frac{1}{4}$ as long. How does the volume of the second box compare to the volume of the first box?
6. Rina’s fish tank has a volume of 8,000 cubic inches. The dimensions of Ava’s fish tank are all $\frac{1}{2}$ the size of Rina’s. What is the volume of Ava’s fish tank?
7. A prism has a width of 6 feet, a length of 8 feet and a height of 12 feet. What is the volume of the prism?
8. What is the area of the base of this prism?
9. What would the volume be of a prism $\frac{1}{4}$ the size of the one describe above?
10. What would the volume be of a prism $\frac{1}{2}$ the size of the one describe above?
11. What would the volume be of a prism twice the size of the one describe above?
12. What ratio can you use to discover the relationship between volume and area?
13. Which measurement will you find when you simplify this ratio?
14. True or false. You can use scale measurement to find the height of a prism.
15. True or false. You can use scale measurement to find the dimensions of a prism.