Problem 90 - Introductory and Intermediate Algebra for College Students 4th

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 is 1000 times larger than its scale model.

Step 1: Calculation of scale factor

The problem imposes that each of the dimensions (length, width, height) of the car model are one tenth of the equivalent dimensions of the real car, implying the scale factor is 1 / 10 or 0.1

Step 2: Understanding volume property

The volume of three-dimensional objects is a property which scales with the cube of the scale factor. This means that you need to cube the scaling factor to compare the volumes.

Step 3: Compute the volume scale

We will now cubing the scale factor to find how many times the actual car's volume is larger than the model's volume: \((\frac{1}{10})^3 = \frac{1}{1000}\). This indicates that the volume of the real car is 1000 times larger than the model.