Answer :
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:
[tex]\[ \text{Ratio of side lengths} = \frac{\text{Side length of larger pentagon}}{\text{Side length of smaller pentagon}} = \frac{10}{6} = \frac{5}{3} \][/tex]
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:
[tex]\[ \text{Area scale factor} = \left( \frac{5}{3} \right)^2 = \frac{25}{9} \approx 2.777777777777778 \][/tex]
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:
[tex]\[ \text{Area of larger pentagon} = \text{Area of smaller pentagon} \times \text{Area scale factor} = 72 \, \text{in}^2 \times 2.777777777777778 = 200 \, \text{in}^2 \][/tex]
Thus, the area of the larger pentagon is approximately [tex]\( 200 \, \text{in}^2 \)[/tex].
So, the area of the larger pentagon is [tex]\( 200 \, \text{in}^2 \)[/tex].
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:
[tex]\[ \text{Ratio of side lengths} = \frac{\text{Side length of larger pentagon}}{\text{Side length of smaller pentagon}} = \frac{10}{6} = \frac{5}{3} \][/tex]
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:
[tex]\[ \text{Area scale factor} = \left( \frac{5}{3} \right)^2 = \frac{25}{9} \approx 2.777777777777778 \][/tex]
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:
[tex]\[ \text{Area of larger pentagon} = \text{Area of smaller pentagon} \times \text{Area scale factor} = 72 \, \text{in}^2 \times 2.777777777777778 = 200 \, \text{in}^2 \][/tex]
Thus, the area of the larger pentagon is approximately [tex]\( 200 \, \text{in}^2 \)[/tex].
So, the area of the larger pentagon is [tex]\( 200 \, \text{in}^2 \)[/tex].
