?. A design is made from a wood panel in the shape of a regular decagon, whose side measures $11$ feet and apothem measures $16$ . $93$ feet. Inside the regular decagon shape there is a hole that is also shaped as a regular decagon, but whose side measures $2$ feet and apothem measures $3$ . $08$ feet. $\newline$ What is the area of the panel of wood without the hole? Round your answer to the nearest hundredth.

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Q. ?. A design is made from a wood panel in the shape of a regular decagon, whose side measures

11

feet and apothem measures

16

93

feet. Inside the regular decagon shape there is a hole that is also shaped as a regular decagon, but whose side measures

2

feet and apothem measures

3

08

feet.

\newline

What is the area of the panel of wood without the hole? Round your answer to the nearest hundredth.

Calculate Perimeter of Larger Decagon: To find the area of the larger decagon, use the formula for the area of a regular polygon: $\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$ . First, calculate the perimeter of the larger decagon by multiplying the length of one side by the number of sides: $\text{Perimeter} = 11 \, \text{feet} \times 10$ .
Calculate Area of Larger Decagon: Perimeter of the larger decagon = $11 \times 10 = 110$ feet.
Calculate Perimeter of Smaller Decagon: Now, calculate the area of the larger decagon using the apothem: $\text{Area} = \frac{1}{2} \times 110 \text{ feet} \times 16.93 \text{ feet}$ .
Calculate Area of Smaller Decagon: Area of the larger decagon = $(\frac{1}{2}) \times 110 \times 16.93 = 930.65$ square feet.
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter = $2 \text{ feet} \times 10$ .
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter = $2$ feet $\times 10$ . Perimeter of the smaller decagon = $2 \times 10 = 20$ feet.
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter = $2 \text{ feet} \times 10$ . Perimeter of the smaller decagon = $2 \times 10 = 20 \text{ feet}$ . Now, calculate the area of the smaller decagon using its apothem: Area = $(1/2) \times 20 \text{ feet} \times 3.08 \text{ feet}$ .
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter = $2 \text{ feet} \times 10$ . Perimeter of the smaller decagon = $2 \times 10 = 20 \text{ feet}$ . Now, calculate the area of the smaller decagon using its apothem: Area = $(1/2) \times 20 \text{ feet} \times 3.08 \text{ feet}$ . Area of the smaller decagon = $(1/2) \times 20 \times 3.08 = 30.8 \text{ square feet}$ .
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter = $2 \text{ feet} \times 10$ . Perimeter of the smaller decagon = $2 \times 10 = 20 \text{ feet}$ . Now, calculate the area of the smaller decagon using its apothem: Area = $(1/2) \times 20 \text{ feet} \times 3.08 \text{ feet}$ . Area of the smaller decagon = $(1/2) \times 20 \times 3.08 = 30.8 \text{ square feet}$ . Finally, subtract the area of the smaller decagon from the area of the larger decagon to find the area of the wood panel without the hole: $930.65 \text{ square feet} - 30.8 \text{ square feet}$ .
Find Area of Wood Panel: Next, calculate the perimeter of the smaller decagon (the hole) in the same way: Perimeter $= 2 \text{ feet} \times 10$ . Perimeter of the smaller decagon $= 2 \times 10 = 20 \text{ feet}$ . Now, calculate the area of the smaller decagon using its apothem: Area $= \frac{1}{2} \times 20 \text{ feet} \times 3.08 \text{ feet}$ . Area of the smaller decagon $= \frac{1}{2} \times 20 \times 3.08 = 30.8 \text{ square feet}$ . Finally, subtract the area of the smaller decagon from the area of the larger decagon to find the area of the wood panel without the hole: $930.65 \text{ square feet} - 30.8 \text{ square feet}$ . Area of the wood panel without the hole $= 930.65 - 30.8 = 899.85 \text{ square feet}$ . Round this to the nearest hundredth.