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The image shows a cone that has a base with an area of square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is square centimeters.
What was the scale factor of the dilation?
Suppose a solid is dilated. If we know the factor by which the surface area or volume scale changed, we can work backward to find the scale factor of dilation. Then we can use that information to solve problems.
A company sells 10 inch by 10 inch by 14 inch 5-gallon aquariums, but a museum wants to buy a 135-gallon aquarium with the same shape. The company needs to know the dimensions of the new tank and by what factor the surface area will change.
Gallons are a measure of volume. So, the volume of the tank increases by a factor of . To find the scale factor for the dimensions of the tank, calculate the cube root of 27, or 3. This tells us that the height, length, and width of the tank will each be multiplied by 3. Next, we can square the scale factor of 3 to find that the tank’s surface area will increase by a factor of 32 = 9.
|
original aquarium |
dilated aquarium |
|
|---|---|---|
| height (inches) | 10 | |
| length (inches) | 14 | |
| width (inches) | 10 | |
| surface area (square inches) | 760 | |
| volume (gallons) | 5 |