Question

a triangle upper A upper B upper C; The edge upper A upper B is labeled c. The edge upper A upper C is labeled b. The edge upper B upper C is labeled a.
If the area of △ABC=134 square feet, m∠C = 41° , and b=24 feet, then what is the length of side a ? Round your answer to the nearest whole number.

(1 point)
Responses

9 feet
9 feet

16 feet
16 feet

17 feet
17 feet

15 feet

Answer 1

To find the length of side \( a \) in triangle \( ABC \) given the area, side \( b \), and angle \( C \), we can use the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)
\]

Here, we know:
- Area = 134 square feet
- \( b = 24 \) feet
- \( \angle C = 41^\circ \)

Plugging the known values into the area formula, we can solve for \( a \):

\[
134 = \frac{1}{2} \times a \times 24 \times \sin(41^\circ)
\]

Now, calculate \(\sin(41^\circ)\):

\[
\sin(41^\circ) \approx 0.6561
\]

Substituting this value into the equation gives:

\[
134 = \frac{1}{2} \times a \times 24 \times 0.6561
\]

This simplifies to:

\[
134 = 12 \times a \times 0.6561
\]

Calculating \( 12 \times 0.6561 \):

\[
12 \times 0.6561 \approx 7.9932
\]

Now, we can substitute this back into the equation:

\[
134 = 7.9932 \times a
\]

To solve for \( a \):

\[
a = \frac{134}{7.9932} \approx 16.76
\]

Rounding this to the nearest whole number, we find:

\[
a \approx 17
\]

Therefore, the length of side \( a \) is \( \boxed{17} \) feet.