Question

If the area of triangle ABC =73 m^2, side b=16 meters, and m angle C= 66 degrees, then what is the length of side a to the nearest whole meter?

22 meters
2 meters
5 meters
10 meters

Answers

GPT-4o mini
To find the length of side \( a \) in triangle \( ABC \), we can use the formula for the area of a triangle given by two sides and the included angle:

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

In this case, we know:

- Area = 73 m²
- Side \( b = 16 \) m
- Angle \( C = 66^\circ \)

We need to find side \( a \). First, we can rearrange the formula to solve for \( a \):

\[
73 = \frac{1}{2} \times a \times 16 \times \sin(66^\circ)
\]

Multiplying both sides by 2:

\[
146 = a \times 16 \times \sin(66^\circ)
\]

Now, we can solve for \( a \):

\[
a = \frac{146}{16 \times \sin(66^\circ)}
\]

Now, we calculate \( \sin(66^\circ) \):

\[
\sin(66^\circ) \approx 0.9135
\]

Substituting \( \sin(66^\circ) \):

\[
a = \frac{146}{16 \times 0.9135}
\]

Calculating the denominator:

\[
16 \times 0.9135 \approx 14.616
\]

Now, substituting back:

\[
a = \frac{146}{14.616} \approx 10.00
\]

Thus, to the nearest whole meter, the length of side \( a \) is approximately:

\[
\boxed{10}
\]

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