If a = 22 ft, b = 34 ft, and c = 37 ft, what is the area of the backyard? A. 1,003 ft2 B. 93 ft2 C. 27,676 ft2 D. 1,258 ft2

1. ### Break Down the Problem

To find the area of the backyard given the side lengths $a$, $b$, and $c$, we determine whether this is a triangle and apply Heron's formula.

2. ### Relevant Concepts

Heron's formula states that the area $A$ of a triangle with side lengths $a$, $b$, and $c$ is given by:

$s = \frac{a + b + c}{2}$ $A = \sqrt{s(s-a)(s-b)(s-c)}$

where $s$ is the semi-perimeter of the triangle.

3. ### Analysis and Detail

1. Calculate the semi-perimeter $s$: $s = \frac{22 + 34 + 37}{2} = \frac{93}{2} = 46.5 \text{ ft}$

2. Calculate each term in Heron’s formula: $s - a = 46.5 - 22 = 24.5 \text{ ft}$ $s - b = 46.5 - 34 = 12.5 \text{ ft}$ $s - c = 46.5 - 37 = 9.5 \text{ ft}$

3. Apply Heron's formula to find the area $A$: $A = \sqrt{46.5 \times 24.5 \times 12.5 \times 9.5}$

4. Calculate the area:

   • First, compute the product: $46.5 \times 24.5 \times 12.5 \times 9.5 \approx 164863.5625$
   • Now, compute the square root: $A \approx \sqrt{164863.5625} \approx 405.99 \text{ ft}^2$

4. ### Verify and Summarize

The area calculated using Heron's formula shows that the area is not directly corresponding to the options given. It suggests that the values may not form a triangle with the area matching the choices listed.

Final Answer

After necessary calculations and assumptions, it appears that the area does not match any of the given options directly. Thus the answer could be derived back with the options available, and given simplicity in the final evaluations:

None of the provided options (A. 1,003 ft², B. 93 ft², C. 27,676 ft², D. 1,258 ft²) seem to represent the area for sides of lengths 22 ft, 34 ft, and 37 ft calculated using Heron's formula.