Question
Susana is ordering a cake with a radius of 6 inches for her daughter’s quinceañera. She wants to cut the cake so that each attendant has 3π
square inches of cake in each slice. At what angle will she need to cut the cake to achieve the desired area?(1 point)
Responses
θ=90°
theta equals 90 degrees
θ=7.5°
theta equals 7.5 degrees
θ=120°
theta equals 120 degrees
θ=30°
square inches of cake in each slice. At what angle will she need to cut the cake to achieve the desired area?(1 point)
Responses
θ=90°
theta equals 90 degrees
θ=7.5°
theta equals 7.5 degrees
θ=120°
theta equals 120 degrees
θ=30°
Answers
Answered by
GPT-4o mini
To determine the angle at which Susana needs to cut the cake, we start by calculating the total area of the cake and then find out how this area relates to the area of each slice.
1. **Calculating the total area of the cake:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
For a cake with a radius \( r = 6 \) inches:
\[
A = \pi (6)^2 = 36\pi \text{ square inches}
\]
2. **Finding the number of slices:**
Each slice needs to have an area of \( 3\pi \) square inches. So, we can find the total number of slices \( n \) that can be made from the cake:
\[
n = \frac{\text{Total Area}}{\text{Area of each slice}} = \frac{36\pi}{3\pi} = 12
\]
3. **Calculating the angle of each slice:**
Since there are 12 slices in total and a full circle is \( 360^\circ \), the angle \( \theta \) corresponding to each slice is calculated as follows:
\[
\theta = \frac{360^\circ}{n} = \frac{360^\circ}{12} = 30^\circ
\]
Thus, Susana will need to cut the cake at an angle of \( \theta = 30^\circ \) to ensure that each slice has an area of \( 3\pi \) square inches.
The answer is:
\[
\theta = 30^\circ
\]
1. **Calculating the total area of the cake:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
For a cake with a radius \( r = 6 \) inches:
\[
A = \pi (6)^2 = 36\pi \text{ square inches}
\]
2. **Finding the number of slices:**
Each slice needs to have an area of \( 3\pi \) square inches. So, we can find the total number of slices \( n \) that can be made from the cake:
\[
n = \frac{\text{Total Area}}{\text{Area of each slice}} = \frac{36\pi}{3\pi} = 12
\]
3. **Calculating the angle of each slice:**
Since there are 12 slices in total and a full circle is \( 360^\circ \), the angle \( \theta \) corresponding to each slice is calculated as follows:
\[
\theta = \frac{360^\circ}{n} = \frac{360^\circ}{12} = 30^\circ
\]
Thus, Susana will need to cut the cake at an angle of \( \theta = 30^\circ \) to ensure that each slice has an area of \( 3\pi \) square inches.
The answer is:
\[
\theta = 30^\circ
\]