Question

a painter needs to find the area of the gable end of a house. what is the area of the gable if it is a triangle with two sides of 81 feet that meet at a 112° angle? ? ft² round to the nearest square foot.

Explanation:

Step1: Recall the formula for the area of a triangle with two sides and included angle

The formula for the area \( A \) of a triangle when two sides \( a \) and \( b \) and the included angle \( C \) are known is \( A=\frac{1}{2}ab\sin C \). Here, \( a = 81 \) feet, \( b = 81 \) feet, and \( C=112^{\circ} \).

Step2: Substitute the values into the formula

Substitute \( a = 81 \), \( b = 81 \), and \( C = 112^{\circ} \) into the formula:
\( A=\frac{1}{2}\times81\times81\times\sin(112^{\circ}) \)
First, calculate \( 81\times81 = 6561 \). Then, \( \frac{1}{2}\times6561=3280.5 \).
Now, find \( \sin(112^{\circ}) \). Using a calculator, \( \sin(112^{\circ})\approx\sin(180^{\circ} - 68^{\circ})=\sin(68^{\circ})\approx0.9272 \).
Then, \( A = 3280.5\times0.9272 \)

Step3: Calculate the final value

\( 3280.5\times0.9272\approx3280.5\times0.9272 = 3041.6796 \)
Rounding to the nearest square foot, we get \( 3042 \).

Answer:

\( 3042 \)