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. (1 point)
Answers
GPT-4o mini
To find the area of the cookie cake that was eaten, we first need to calculate the total area of the cookie cake and then find the area corresponding to the 270-degree angle that was eaten.
1. **Calculate the radius of the cookie cake:**
The diameter of the cookie cake is 16 inches, so the radius \( r \) is:
\[
r = \frac{diameter}{2} = \frac{16}{2} = 8 \text{ inches}
\]
2. **Calculate the total area of the cookie cake:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Substituting the radius:
\[
A = \pi (8^2) = \pi (64) = 64\pi
\]
3. **Calculate the area of the eaten section:**
The angle of the eaten section is 270 degrees. The fraction of the circle that this angle represents can be calculated as:
\[
\text{Fraction of circle} = \frac{270}{360} = \frac{3}{4}
\]
Therefore, the area of the eaten section \( A_{eaten} \) is:
\[
A_{eaten} = \text{Fraction of circle} \times \text{Total area} = \frac{3}{4} \times 64\pi
\]
Simplifying this gives:
\[
A_{eaten} = 48\pi
\]
4. **Calculate the numerical value of the eaten area:**
Using \( \pi \approx 3.14 \):
\[
A_{eaten} \approx 48 \times 3.14 = 150.72
\]
5. **Round the answer to the nearest tenth:**
Rounding 150.72 to the nearest tenth gives:
\[
A_{eaten} \approx 150.7 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.
1. **Calculate the radius of the cookie cake:**
The diameter of the cookie cake is 16 inches, so the radius \( r \) is:
\[
r = \frac{diameter}{2} = \frac{16}{2} = 8 \text{ inches}
\]
2. **Calculate the total area of the cookie cake:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Substituting the radius:
\[
A = \pi (8^2) = \pi (64) = 64\pi
\]
3. **Calculate the area of the eaten section:**
The angle of the eaten section is 270 degrees. The fraction of the circle that this angle represents can be calculated as:
\[
\text{Fraction of circle} = \frac{270}{360} = \frac{3}{4}
\]
Therefore, the area of the eaten section \( A_{eaten} \) is:
\[
A_{eaten} = \text{Fraction of circle} \times \text{Total area} = \frac{3}{4} \times 64\pi
\]
Simplifying this gives:
\[
A_{eaten} = 48\pi
\]
4. **Calculate the numerical value of the eaten area:**
Using \( \pi \approx 3.14 \):
\[
A_{eaten} \approx 48 \times 3.14 = 150.72
\]
5. **Round the answer to the nearest tenth:**
Rounding 150.72 to the nearest tenth gives:
\[
A_{eaten} \approx 150.7 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.