Question

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 15 feet

Answer 1

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

\[
\text{Area} = \frac{1}{2}ab \sin(C)
\]

where \( a \) and \( b \) are the lengths of two sides of the triangle and \( C \) is the angle between them.

In this case, we know:

- Area \( = 134 \) square feet
- \( b = 24 \) feet
- \( C = 41^\circ \)

Substituting the known values into the area formula gives:

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

To isolate \( a \), we first simplify the equation. The sine of \( 41^\circ \) is approximately \( 0.6561 \) (use a calculator). Thus, we calculate:

\[
134 = 12a \cdot \sin(41^\circ)
\]
\[
134 = 12a \cdot 0.6561
\]
\[
134 = 7.8732a
\]

Now, solve for \( a \):

\[
a = \frac{134}{7.8732} \approx 17.01
\]

Rounding to the nearest whole number gives:

\[
a \approx 17
\]

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