McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
6. Dilations
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Exercise 31 Page 679

Practice makes perfect
a We want to find the surface area and volume of the prism shown below.

Surface Area of the Prism

The surface area of a prism can be found using the formula below.
In the formula, is the perimeter of the base, is the area of the base and is the height of the prism. Here the base is wide and long. We can use these values to calculate the perimeter and area of the base.
Now, let's substitute and into the formula for the surface area of a prism.
The surface area of the prism is

Volume of the Prism

To find the volume of the prism, we multiply the area of the base by the height
We know that and Let's substitute these values into the formula and simplify.
The volume of the prism is
b Our goal is to find the surface area and the volume of the prism after a dilation with a scale factor of See that this type of a dilation makes each dimension twice as long. Let's find the dimensions of the image.
Preimage Image
The base of the image is wide and long.
Let's substitute and into the formulas for the surface area and the volume of a prism and simplify.
Surface Area Volume
Formula
Substitution
Calculation
c This time we want to find the surface area and the volume of the prism after a dilation with a scale factor of Let's find the dimensions of the image.
Preimage Image
The base of the image is wide and long. Let's find the perimeter and area of the base.
We can now substitute and into the formulas for the surface area and the volume of a prism.
Surface Area Volume,
Formula
Substitution
Calculation
d We want to compare the perimeters and areas of the image and preimage in each dilation. We will start by comparing the surface areas of the preimage, Image B, and Image C.

Comparing Surface Areas

Let's make a table showing the surface areas and volumes of the given prism and the two images.

Preimage Image B Image C
Surface Area
Volume

We see that a dilation with a scale factor of makes the surface area times more. Also, a dilation with a scale factor of makes the surface area of what it used to be.

Preimage Image B Image C
Surface Area
Volume

Comparing Volumes

Now we can compare the volumes.

Preimage Image B Image C
Surface Area
Volume

A dilation with a scale factor of makes the volume times more. A dilation with a scale factor of makes the volume of what it used to be.

e We want to decide on the effect that a dilation by a scale factor has on the surface area and volume of a prism. For that let's remember the table from Part D.
Preimage Image B Image C
Surface Area
Volume

See that with a dilation of the surface area was multiplied by and the volume was multiplied by With a dilation of the surface area was multiplied by and the volume was multiplied by

Preimage Image B Image C
Surface Area
Volume

We can conclude that if is the scale factor of a dilation of a prism, the surface area of the preimage would be multiplied by and the volume of the preimage would be multiplied by