Question

a cookie cake has a diameter of 18 inches if part of the cookie cake is eaten, the missing section forms an angle of 270 degrees. what is the rea of the cookie cake that was eaten? round your answer to the nearest tenth

Answer 1

To find the area of the cookie cake that was eaten, we first calculate the area of the entire cookie cake and then determine the area corresponding to the missing 270-degree section.

1. **Calculate the total area of the cookie cake**:
The formula for the area \(A\) of a circle is given by:
\[
A = \pi r^2
\]
where \(r\) is the radius. The diameter of the cookie cake is 18 inches, so the radius \(r\) is:
\[
r = \frac{18}{2} = 9 \text{ inches}
\]
Now we can calculate the area:
\[
A = \pi (9^2) = \pi (81) = 81\pi \text{ square inches}
\]

2. **Calculate the proportion of the cookie cake corresponding to the 270-degree angle**:
A full circle is 360 degrees. The fraction of the circle that corresponds to the 270 degrees eaten is:
\[
\frac{270}{360} = \frac{3}{4}
\]

3. **Calculate the area that was eaten**:
To find the area that corresponds to this fraction, we multiply the total area by \(\frac{3}{4}\):
\[
\text{Area eaten} = \frac{3}{4} \times 81\pi
\]
Calculating this gives:
\[
\text{Area eaten} = \frac{243}{4}\pi
\]

4. **Finding the numerical value**:
Using \(\pi \approx 3.14\),
\[
\text{Area eaten} \approx \frac{243}{4} \times 3.14 \approx 60.75 \times 3.14 \approx 190.35
\]

Rounding this to the nearest tenth, the area of the cookie cake that was eaten is approximately:
\[
\boxed{190.4} \text{ square inches}
\]