To determine the correct way to calculate the area of the stop sign, let's walk through the process step-by-step.
### Step 1: Understand the Shape and Dimensions
Given:
- The stop sign is a regular octagon.
- The width and height of the octagon are both 30 inches.
### Step 2: Calculate the Side Length of the Regular Octagon
The width (or height) of a regular octagon relates to the side length and a geometric constant. The formula relating the width to the side length [tex]\( a \)[/tex] is:
[tex]\[ \text{width} = 2a(1 + \sqrt{2}) \][/tex]
Given the width is 30 inches, we can solve for the side length [tex]\( a \)[/tex]:
[tex]\[ 30 = 2a(1 + \sqrt{2}) \implies a = \frac{30}{2(1 + \sqrt{2})} \][/tex]
Using the calculated result, we get:
[tex]\[ a \approx 6.213 \text{ inches} \][/tex]
### Step 3: Calculate the Perimeter of the Octagon
The perimeter [tex]\( P \)[/tex] of a regular octagon is given by multiplying the side length [tex]\( a \)[/tex] by 8:
[tex]\[ P = 8a \][/tex]
Using the calculated side length:
[tex]\[ P = 8 \times 6.213 \approx 49.706 \text{ inches} \][/tex]
### Step 4: Calculate the Apothem of the Octagon
The apothem [tex]\( a_p \)[/tex] of a regular octagon can be calculated using the side length [tex]\( s \)[/tex]:
[tex]\[ a_p = \frac{s}{2 \tan(\pi/8)} \][/tex]
Using the side length:
[tex]\[ a_p \approx 7.5 \text{ inches} \][/tex]
### Step 5: Calculate the Area of the Octagon
The area [tex]\( A \)[/tex] of a regular octagon is given by:
[tex]\[ A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} \][/tex]
Using the calculated values:
[tex]\[ A = \frac{1}{2} \times 49.706 \times 7.5 \approx 186.396 \text{ square inches} \][/tex]
### Conclusion
Looking at the options provided:
1. [tex]\( A = (B)(H) \Rightarrow A \approx 900 \text{ square inches} \)[/tex]
2. [tex]\( A = \frac{1}{2}(B)(H) \Rightarrow A \approx 225 \text{ square inches} \)[/tex]
3. [tex]\( A = \frac{1}{2}(a)(P) \Rightarrow A \approx 1491 \text{ square inches} \)[/tex]
4. [tex]\( A = \frac{1}{2}(a)(P) \cos A \approx 746 \text{ square inches} \)[/tex]
None of these options perfectly match [tex]\( 186.396 \)[/tex] square inches exactly, which was determined through detailed calculation. However, the method for the closest more elegant approximation involves using [tex]\( A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} \)[/tex], so none of the options provided exactly align with the step-by-step solution done here, but this should be the ideal method to compute the area of the octagon.
