Juan drew a circle with a radius of 4 cm. He drew two radii that formed a 150° angle. He used the steps below to find the area of the sector formed by the 150° angle.

1. Area of Circle: [tex]\( A = \pi r^2 = \pi (4^2) = 16 \pi \)[/tex]

2. Area of Sector:
[tex]\[
\frac{150}{360} \times 16 \pi = \frac{5}{12} \times 16 \pi = \frac{80 \pi}{12} = \frac{20 \pi}{3} \approx 6.67 \pi
\][/tex]

What error did Juan make in finding that the area of the sector is [tex]\( 38.4 \pi \)[/tex] cm²?
A. He found an incorrect area for the entire circle.
B. He set up the proportion incorrectly.
C. He solved the proportion incorrectly.
D. He used the area of the circle instead of the circumference.



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.