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Chapter 1: Problem 1

Two rugs have exactly the same shape, but one is twice as long as the other. Does that mean its area is also twice as large? Explain.

Short Answer

Expert verified
Answer: No, the area of Rug 2 is actually four times the area of Rug 1, not twice as large.

Step by step solution

01

Write down the given information.

Let Rug 1 be the smaller rug, and Rug 2 be the larger rug which is twice as long. We are given that Rug 2 is similar to Rug 1, and the length of Rug 2 is twice the length of Rug 1.
02

Choose a variable for the side length.

Let's use "s" to represent the side length of Rug 1, which means the side length of Rug 2 is "2s".
03

Determine the dimensions of the rugs.

Since the rugs have the same shape, we can assume that they're both rectangles. If the length of Rug 1 is "s" and Rug 2 is "2s", we can assume that Rug 1 has dimensions "s" by "x" and Rug 2 has dimensions "2s" by "y", with x and y being the unknown widths of the rugs.
04

Find the ratio of the widths.

As the rugs are similar, the ratio of their corresponding widths will be the same as the ratio of their lengths. So, the ratio of the width of Rug 1 to Rug 2 will also be 1:2, which means y=2x.
05

Calculate the area of the rugs.

Now we can calculate the area of Rug 1 (A1) and Rug 2 (A2). The area of a rectangle is given by the formula: Area = length × width. So: A1 = s × x A2 = 2s × (2x)
06

Compare the areas to check if they are twice as large.

Now let's find out if the area of Rug 2 is twice the area of Rug 1. A2 = 2s × (2x) = 4(sx) Compare A2 to A1: A1 = s × x A2 = 4(s × x) = 4A1
07

Write the conclusion.

The area of Rug 2 (A2) is four times the area of Rug 1 (A1), not twice as large as Rug 1. Therefore, just because the length of one rug is twice as long as the other, it does not mean its area is also twice as large.

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