Questions: Emily has been hired to create a cake with a diameter of 34 feet for a special event. However, the client has requested that one of the cake layers be cut into a specific shape to match the theme of the event. The client has requested that Emily cut the layer into a sector of a circle with an angle of 4 pi/3 radians. Emily needs to calculate the area of the sector to determine how much cake batter she will need to make for that layer. Round your answer to four decimal places. A= Number ft^2

Emily has been hired to create a cake with a diameter of 34 feet for a special event.
However, the client has requested that one of the cake layers be cut into a specific shape to match the theme of the event. The client has requested that Emily cut the layer into a sector of a circle with an angle of 4 pi/3 radians. Emily needs to calculate the area of the sector to determine how much cake batter she will need to make for that layer.

Round your answer to four decimal places.
A= Number ft^2

Transcript text: Emily has been hired to create a cake with a diameter of 34 feet for a special event. However, the client has requested that one of the cake layers be cut into a specific shape to match the theme of the event. The client has requested that Emily cut the layer into a sector of a circle with an angle of $\frac{4 \pi}{3}$ radians. Emily needs to calculate the area of the sector to determine how much cake batter she will need to make for that layer. Round your answer to four decimal places. \[ A=\text { Number } \mathrm{ft}^{2} \] Show your work and explain, in your own words, how you arrived at your answer. Answers with no relevant explanations may receive reduced or no credit.

Solution

Solution Steps

To find the area of a sector of a circle, we can use the formula for the area of a sector, which is given by:

\[ A = \frac{1}{2} \times r^2 \times \theta \]

where $ r $ is the radius of the circle and $ \theta $ is the angle in radians. Given the diameter of the cake is 34 feet, the radius $ r $ is half of the diameter. The angle $ \theta $ is given as $ \frac{4 \pi}{3} $ radians. We will plug these values into the formula to find the area of the sector.

Step 1: Calculate the Radius

Given the diameter of the cake is $ 34 $ feet, we can find the radius $ r $ using the formula: \[ r = \frac{d}{2} = \frac{34}{2} = 17 \text{ feet} \]

Step 2: Identify the Angle

The angle $ \theta $ for the sector is given as: \[ \theta = \frac{4\pi}{3} \text{ radians} \]

Step 3: Calculate the Area of the Sector

Using the formula for the area $ A $ of a sector: \[ A = \frac{1}{2} r^2 \theta \] we substitute $ r = 17 $ and $ \theta = \frac{4\pi}{3} $: \[ A = \frac{1}{2} \times (17)^2 \times \frac{4\pi}{3} \] Calculating this gives: \[ A = \frac{1}{2} \times 289 \times \frac{4\pi}{3} = \frac{1156\pi}{6} \approx 605.2802 \text{ ft}^2 \]