Problem 90 - Introductory Algebra for College Students

Question

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?

Step-by-Step Solution

Answer

The volume of the car exceeds the scale model by 1000 times.

Step 1: 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.

Step 2: 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.

Step 3: 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 \(L × W × H\). Now, for the scale model, every dimension is \(\frac{1}{10}\) of the original. Therefore, the model car's volume will be \(\left(\frac{L}{10}\right) × \left(\frac{W}{10}\right) × \left(\frac{H}{10}\right)\)

Step 4: 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: \(\frac{L × W × H}{\left(\frac{L}{10}\right) × \left(\frac{W}{10}\right) × \left(\frac{H}{10}\right)} = \frac{1}{\left(\frac{1}{10}\right)^3} = 1000\). This indicates that the original car's volume is 1000 times that of the scale model.