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1. Suppose there are two regular pentagons that are similar. The smaller one has five sides with a length of
6 in and the larger one has five sides with a length of 10 in. The area of the smaller pentagon is 72 in².
Find the area of the larger pentagon. (Hint: use area scale factor to solve the problem more quickly).

Verified Answer :

GinnyAnswer

Certainly! Let's solve the problem step-by-step.

1. Understand the problem:
We have two similar regular pentagons. The smaller pentagon has a side length of 6 inches and an area of 72 square inches. The larger pentagon has a side length of 10 inches. We are asked to find the area of the larger pentagon.

2. Identify the side lengths of the pentagons:
- Smaller pentagon: side length = 6 inches
- Larger pentagon: side length = 10 inches

3. Calculate the ratio of the side lengths:
\(\[ \text{Ratio of side lengths} = \frac{\text{Side length of larger pentagon}}{\text{Side length of smaller pentagon}} = \frac{10}{6} = \frac{5}{3} \]\)

4. Determine the area scale factor:
Since the pentagons are similar, the ratio of their areas is the square of the ratio of their side lengths. Therefore:
\(\[ \text{Area scale factor} = \left( \frac{5}{3} \right)^2 = \frac{25}{9} \approx 2.777777777777778 \]\)

5. Calculate the area of the larger pentagon:
The area of the larger pentagon can be found by multiplying the area of the smaller pentagon by the area scale factor:
\(\[ \text{Area of larger pentagon} = \text{Area of smaller pentagon} \times \text{Area scale factor} = 72 \, \text{in}^2 \times 2.777777777777778 = 200 \, \text{in}^2 \]\)

Thus, the area of the larger pentagon is approximately \(\( 200 \, \text{in}^2 \)\).

So, the area of the larger pentagon is \(\( 200 \, \text{in}^2 \)\).

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