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) area = in2
Answers
GPT-4o mini
To find the area of the cookie cake that was eaten, we first need to determine the area of the entire cookie cake and then calculate the fraction of that area corresponding to the missing section (which forms an angle of 270 degrees).
1. **Calculate the Area of the Entire Cookie Cake:**
The formula for the area \(A\) of a circle is given by:
\[
A = \pi r^2
\]
The radius \(r\) of the cookie cake is half of the diameter. Since the diameter is given as 16 inches, the radius is:
\[
r = \frac{16}{2} = 8 \text{ inches}
\]
Now plug the radius into the area formula:
\[
A = \pi (8)^2 = \pi \times 64 = 64\pi \text{ square inches}
\]
Approximating \(\pi\) as 3.14, we calculate:
\[
A \approx 64 \times 3.14 = 200.96 \text{ square inches}
\]
2. **Calculate the Fraction of the Area Corresponding to 270 Degrees:**
A full circle is 360 degrees. The fraction of the circle representing the 270 degrees that was eaten is:
\[
\text{Fraction} = \frac{270}{360} = \frac{3}{4}
\]
3. **Calculate the Area of the Section that was Eaten:**
Now, to find the area that was eaten, we multiply the total area of the cookie cake by the fraction we just calculated:
\[
\text{Area eaten} = \frac{3}{4} \times A \approx \frac{3}{4} \times 200.96
\]
Calculating that gives:
\[
\text{Area eaten} \approx 0.75 \times 200.96 = 150.72 \text{ square inches}
\]
4. **Round the Area to the Nearest Tenth:**
Rounding 150.72 to the nearest tenth, we get:
\[
\text{Area eaten} \approx 150.7 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.
1. **Calculate the Area of the Entire Cookie Cake:**
The formula for the area \(A\) of a circle is given by:
\[
A = \pi r^2
\]
The radius \(r\) of the cookie cake is half of the diameter. Since the diameter is given as 16 inches, the radius is:
\[
r = \frac{16}{2} = 8 \text{ inches}
\]
Now plug the radius into the area formula:
\[
A = \pi (8)^2 = \pi \times 64 = 64\pi \text{ square inches}
\]
Approximating \(\pi\) as 3.14, we calculate:
\[
A \approx 64 \times 3.14 = 200.96 \text{ square inches}
\]
2. **Calculate the Fraction of the Area Corresponding to 270 Degrees:**
A full circle is 360 degrees. The fraction of the circle representing the 270 degrees that was eaten is:
\[
\text{Fraction} = \frac{270}{360} = \frac{3}{4}
\]
3. **Calculate the Area of the Section that was Eaten:**
Now, to find the area that was eaten, we multiply the total area of the cookie cake by the fraction we just calculated:
\[
\text{Area eaten} = \frac{3}{4} \times A \approx \frac{3}{4} \times 200.96
\]
Calculating that gives:
\[
\text{Area eaten} \approx 0.75 \times 200.96 = 150.72 \text{ square inches}
\]
4. **Round the Area to the Nearest Tenth:**
Rounding 150.72 to the nearest tenth, we get:
\[
\text{Area eaten} \approx 150.7 \text{ square inches}
\]
Thus, the area of the cookie cake that was eaten is approximately \( \boxed{150.7} \) square inches.