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 determine what portion of that area corresponds to the 270 degrees that were eaten.
1. **Calculate the total area of the cookie cake**:
The formula for the area of a circle is given by:
\[
A = \pi r^2
\]
The diameter of the cookie cake is 16 inches, so the radius \( r \) is half of the diameter:
\[
r = \frac{16}{2} = 8 \text{ inches}
\]
Now, we can calculate the total area:
\[
A = \pi (8^2) = \pi \times 64 \approx 201.06 \text{ square inches} \quad (\text{using } \pi \approx 3.14)
\]
2. **Determine the fraction of the cake that corresponds to the eaten section**:
The total angle of a circle is 360 degrees. The eaten section has an angle of 270 degrees, so the fraction of the circle that was eaten is:
\[
\text{Fraction eaten} = \frac{270}{360} = \frac{3}{4}
\]
3. **Calculate the area of the eaten section**:
To find the area of the eaten section, we multiply the total area by the fraction that was eaten:
\[
\text{Area eaten} = A \times \frac{3}{4} = 201.06 \times \frac{3}{4} \approx 201.06 \times 0.75 \approx 150.795
\]
4. **Round to the nearest tenth**:
Rounding \( 150.795 \) to the nearest tenth gives us:
\[
\text{Area eaten} \approx 150.8 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.8} \) square inches.
1. **Calculate the total area of the cookie cake**:
The formula for the area of a circle is given by:
\[
A = \pi r^2
\]
The diameter of the cookie cake is 16 inches, so the radius \( r \) is half of the diameter:
\[
r = \frac{16}{2} = 8 \text{ inches}
\]
Now, we can calculate the total area:
\[
A = \pi (8^2) = \pi \times 64 \approx 201.06 \text{ square inches} \quad (\text{using } \pi \approx 3.14)
\]
2. **Determine the fraction of the cake that corresponds to the eaten section**:
The total angle of a circle is 360 degrees. The eaten section has an angle of 270 degrees, so the fraction of the circle that was eaten is:
\[
\text{Fraction eaten} = \frac{270}{360} = \frac{3}{4}
\]
3. **Calculate the area of the eaten section**:
To find the area of the eaten section, we multiply the total area by the fraction that was eaten:
\[
\text{Area eaten} = A \times \frac{3}{4} = 201.06 \times \frac{3}{4} \approx 201.06 \times 0.75 \approx 150.795
\]
4. **Round to the nearest tenth**:
Rounding \( 150.795 \) to the nearest tenth gives us:
\[
\text{Area eaten} \approx 150.8 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.8} \) square inches.