Answer :
Let's analyze Juan's steps to identify the error he made.
First, let's calculate the correct area of the circle.
1. Find the area of the circle:
- The formula for the area of a circle with radius [tex]\( r \)[/tex] is [tex]\( A = \pi r^2 \)[/tex].
- Given [tex]\( r = 4 \)[/tex] cm, substitute this into the formula:
[tex]\[ A = \pi (4)^2 = 16\pi \text{ cm}^2 \][/tex]
Next, let's determine the correct area of the sector.
2. Find the area of the sector:
- The formula for the area of a sector of a circle with a central angle [tex]\( \theta \)[/tex] is [tex]\( \text{Area of sector} = \frac{\theta}{360} \times \text{Area of the circle} \)[/tex].
- Given [tex]\( \theta = 150^\circ \)[/tex] and using the area of the circle [tex]\( 16\pi \)[/tex] cm[tex]\(^2\)[/tex]:
[tex]\[ \text{Area of sector} = \frac{150}{360} \times 16\pi \][/tex]
- Simplify the fraction:
[tex]\[ \frac{150}{360} = \frac{5}{12} \][/tex]
- Thus:
[tex]\[ \text{Area of sector} = \frac{5}{12} \times 16\pi = \frac{80\pi}{12} = \frac{20\pi}{3} \approx 20.94\ \text{cm}^2 \][/tex]
We now analyze the steps Juan took to find the area of the sector.
3. Analyze Juan's approach:
- Juan first found the area of the circle correctly: [tex]\( 16\pi \)[/tex] cm[tex]\(^2\)[/tex].
- Juan incorrectly formed the proportion [tex]\(\frac{16\pi}{a} = \frac{150}{360}\)[/tex]. This is not correct for finding the area of the sector.
Here, Juan tried to solve the incorrect equation [tex]\(\frac{16\pi}{a} = \frac{150}{360}\)[/tex]:
[tex]\[ 150a = 5760\pi \][/tex]
[tex]\[ a = 38.4\pi \][/tex]
Error Identification:
- The correct proportion to use should have been setting up the fraction of the angle to the full circle's area directly, i.e., [tex]\(\frac{\theta}{360} \times \text{area of the circle}\)[/tex].
- Thus, Juan made an error by incorrectly setting up the proportion.
Therefore, the correct identification of Juan’s mistake is:
He solved the proportion incorrectly.
First, let's calculate the correct area of the circle.
1. Find the area of the circle:
- The formula for the area of a circle with radius [tex]\( r \)[/tex] is [tex]\( A = \pi r^2 \)[/tex].
- Given [tex]\( r = 4 \)[/tex] cm, substitute this into the formula:
[tex]\[ A = \pi (4)^2 = 16\pi \text{ cm}^2 \][/tex]
Next, let's determine the correct area of the sector.
2. Find the area of the sector:
- The formula for the area of a sector of a circle with a central angle [tex]\( \theta \)[/tex] is [tex]\( \text{Area of sector} = \frac{\theta}{360} \times \text{Area of the circle} \)[/tex].
- Given [tex]\( \theta = 150^\circ \)[/tex] and using the area of the circle [tex]\( 16\pi \)[/tex] cm[tex]\(^2\)[/tex]:
[tex]\[ \text{Area of sector} = \frac{150}{360} \times 16\pi \][/tex]
- Simplify the fraction:
[tex]\[ \frac{150}{360} = \frac{5}{12} \][/tex]
- Thus:
[tex]\[ \text{Area of sector} = \frac{5}{12} \times 16\pi = \frac{80\pi}{12} = \frac{20\pi}{3} \approx 20.94\ \text{cm}^2 \][/tex]
We now analyze the steps Juan took to find the area of the sector.
3. Analyze Juan's approach:
- Juan first found the area of the circle correctly: [tex]\( 16\pi \)[/tex] cm[tex]\(^2\)[/tex].
- Juan incorrectly formed the proportion [tex]\(\frac{16\pi}{a} = \frac{150}{360}\)[/tex]. This is not correct for finding the area of the sector.
Here, Juan tried to solve the incorrect equation [tex]\(\frac{16\pi}{a} = \frac{150}{360}\)[/tex]:
[tex]\[ 150a = 5760\pi \][/tex]
[tex]\[ a = 38.4\pi \][/tex]
Error Identification:
- The correct proportion to use should have been setting up the fraction of the angle to the full circle's area directly, i.e., [tex]\(\frac{\theta}{360} \times \text{area of the circle}\)[/tex].
- Thus, Juan made an error by incorrectly setting up the proportion.
Therefore, the correct identification of Juan’s mistake is:
He solved the proportion incorrectly.
