To find the area of a sector, the formula used is:
\[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius.
For a radius of \( 23 \) meters and \( \theta = 200^\circ \), the area of the sector would be calculated as follows:
\[ \text{Area} = \frac{200}{360} \times \pi \times (23)^2 \]
Calculating \( (23)^2 \):
\[ 23^2 = 529 \]
So plug that into the area formula:
\[ \text{Area} = \frac{200}{360} \times \pi \times 529 \]
Now, simplify \( \frac{200}{360} \):
\[ \frac{200}{360} = \frac{5}{9} \]
Now substitute back into the area formula:
\[ \text{Area} = \frac{5}{9} \times \pi \times 529 \]
Now, approximately calculating:
\[ \frac{5}{9} \times 529 \approx 294.44 \]
Then multiply by \( \pi \) (approximately \( 3.14 \)):
\[ \text{Area} \approx 294.44 \times 3.14 \approx 924.59 , \text{m}^2 \]
As calculated, the correct area is much larger than Danae's answer of \( 40.14 , \text{m}^2 \).
Now let's analyze the statements:
-
Statement #1: She used the diameter instead of the radius.
- This statement does not apply because she used the correct radius value.
-
Statement #2: She incorrectly calculated \( \frac{200}{360} \).
- It's possible she made an error when calculating the fraction, but without knowing her specific calculation, we cannot assume this is the mistake.
-
Statement #3: She forgot to square the radius.
- If Danae did not square the radius in her calculation, this would lead to a significant underestimation of the area, which aligns with her result of \( 40.14 , \text{m}^2 \).
Since the area calculation requires squaring the radius, and if she forgot this step, it explains why the area she calculated was incorrectly low.
The most accurate conclusion about her mistake is:
Statement #3 describes Danae's mistake.