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Chapter 10: Problem 56

A scale model of a car is constructed so that its length, width, and height are each \(\frac{1}{10}\) the length, width, and height of the actual car. By how many times does the volume of the car exceed its scale model?

Short Answer

Expert verified
The volume of the actual car exceeds the volume of its scale model by 1000 times.

Step by step solution

01

Understand the Scale Factor

The scale model of the car is \(\frac{1}{10}\) times the actual size. Hence, the scale factor for the model is \(\frac{1}{10}\) for each of the three dimensions - length, width, and height.
02

Calculate the Volume Ratio

To find out how much the volume of the car exceeds its model, we need to calculate the ratio of the volumes of the actual car to the scale model. The volume ratio would be the cube of the scale factor, as volume is a 3-dimensional property. Therefore, the volume ratio is \(\left(\frac{1}{10}\right)^3\).
03

Invert The Volume Ratio

The scale factor gives us the ratio of the scale model to the actual car. But we need to find how many times the actual car's volume exceeds the model's volume. For that, we would need to invert the volume ratio. Hence, the actual car's volume exceeds the model's volume by \(\frac{1} {(\frac{1}{10})^3}\) times.

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