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Chapter 2: Problem 90

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 car exceeds the scale model by 1000 times.

Step by step solution

01

Understanding the problem

We have a scale model of a car where dimensions length, width, and height are each \(\frac{1}{10}\) of the actual car dimensions. We have to find out by how many times the volume of the original car exceeds that of its scale model.
02

Identify volume of a cuboid

The volume of a cuboid (the car can be considered as a cuboid for simplicity without loss of generality), is given by the formula Volume = Length x Width x Height.
03

Indicate the volume for the original and scaled model

Let's assume the dimensions of the original car to be L for length, W for width and H for height. Therefore, its volume will be \(L 脳 W 脳 H\). Now, for the scale model, every dimension is \(\frac{1}{10}\) of the original. Therefore, the model car's volume will be \(\left(\frac{L}{10}\right) 脳 \left(\frac{W}{10}\right) 脳 \left(\frac{H}{10}\right)\)
04

Calculate the ratio of the volumes

To find out by how much the original car's volume exceeds the model's volume, we take the ratio: \(\frac{L 脳 W 脳 H}{\left(\frac{L}{10}\right) 脳 \left(\frac{W}{10}\right) 脳 \left(\frac{H}{10}\right)} = \frac{1}{\left(\frac{1}{10}\right)^3} = 1000\). This indicates that the original car's volume is 1000 times that of the scale model.

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