Question

of the area of triangle ABC = 134 square feet, m angle C = 41 degrees and b=24 feet, then what is the length of side a? round your answer to the nearest whole number

Answer 1

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

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

Where:
- \( \text{Area} \) is the area of the triangle,
- \( a \) and \( b \) are the lengths of two sides,
- \( C \) is the angle between those two sides.

In this case:
- Area = \( 134 \) square feet
- \( C = 41^\circ \)
- \( b = 24 \) feet

We can rearrange the formula to solve for \( a \):

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

Substituting the known values into the formula:

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

First, we need to calculate \( \sin(41^\circ) \):

\[
\sin(41^\circ) \approx 0.6561 \quad (\text{using a calculator})
\]

Now substitute the value of \( \sin(41^\circ) \):

\[
a = \frac{2 \cdot 134}{24 \cdot 0.6561}
\]
\[
a = \frac{268}{24 \cdot 0.6561}
\]
\[
= \frac{268}{15.7464} \quad (\text{calculating } 24 \cdot 0.6561)
\]
\[
\approx 17.00
\]

Rounding to the nearest whole number:

\[
a \approx 17
\]

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