Question
Sarah's mom bought her a cookie cake for her birthday. It had a diameter of 16 inches. Sarah's little brother ate part of the cookie cake. The missing section forms an angle of 270 degrees. What is the area of the cookie cake that was eaten? Round your answer to the nearest tenth.
Answers
GPT-4o mini
To find the area of the section of the cookie cake that was eaten, we first need to calculate the area of the entire cookie cake and then determine what fraction of that area corresponds to the 270-degree section.
1. **Calculate the radius of the cookie cake:**
\[
\text{Diameter} = 16 \text{ inches} \quad \Rightarrow \quad \text{Radius} = \frac{16}{2} = 8 \text{ inches}
\]
2. **Calculate the area of the entire cookie cake:**
The area \( A \) of a circle can be calculated using the formula:
\[
A = \pi r^2
\]
Substituting the radius:
\[
A = \pi (8^2) = \pi \cdot 64 = 64\pi
\]
3. **Calculate the fraction of the cake that corresponds to the 270-degree section:**
The total angle in a circle is 360 degrees. The fraction of the area that the 270-degree section represents is:
\[
\frac{270}{360} = \frac{3}{4}
\]
4. **Calculate the area of the eaten section:**
The area of the eaten section \( A_{\text{eaten}} \) can be found by multiplying the total area by the fraction:
\[
A_{\text{eaten}} = \frac{3}{4} \cdot A = \frac{3}{4} \cdot 64\pi = 48\pi
\]
5. **Calculate the numerical value of the area:**
Using \( \pi \approx 3.14 \):
\[
A_{\text{eaten}} \approx 48 \cdot 3.14 = 150.72
\]
6. **Round the answer to the nearest tenth:**
\[
A_{\text{eaten}} \approx 150.7
\]
Therefore, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.
1. **Calculate the radius of the cookie cake:**
\[
\text{Diameter} = 16 \text{ inches} \quad \Rightarrow \quad \text{Radius} = \frac{16}{2} = 8 \text{ inches}
\]
2. **Calculate the area of the entire cookie cake:**
The area \( A \) of a circle can be calculated using the formula:
\[
A = \pi r^2
\]
Substituting the radius:
\[
A = \pi (8^2) = \pi \cdot 64 = 64\pi
\]
3. **Calculate the fraction of the cake that corresponds to the 270-degree section:**
The total angle in a circle is 360 degrees. The fraction of the area that the 270-degree section represents is:
\[
\frac{270}{360} = \frac{3}{4}
\]
4. **Calculate the area of the eaten section:**
The area of the eaten section \( A_{\text{eaten}} \) can be found by multiplying the total area by the fraction:
\[
A_{\text{eaten}} = \frac{3}{4} \cdot A = \frac{3}{4} \cdot 64\pi = 48\pi
\]
5. **Calculate the numerical value of the area:**
Using \( \pi \approx 3.14 \):
\[
A_{\text{eaten}} \approx 48 \cdot 3.14 = 150.72
\]
6. **Round the answer to the nearest tenth:**
\[
A_{\text{eaten}} \approx 150.7
\]
Therefore, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.