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 42 feet that meet at a 105° 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, \( a = 42 \) feet, \( b = 42 \) feet, and \( C=105^{\circ} \).

Step2: Substitute the values into the formula

Substitute \( a = 42 \), \( b = 42 \), and \( C = 105^{\circ} \) into the formula:
\( A=\frac{1}{2}\times42\times42\times\sin(105^{\circ}) \)
First, calculate \( 42\times42 = 1764 \). Then, \( \frac{1}{2}\times1764=882 \). Now, we need to find \( \sin(105^{\circ}) \). We know that \( \sin(105^{\circ})=\sin(60^{\circ} + 45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx\frac{2.449 + 1.414}{4}=\frac{3.863}{4}\approx0.9659 \)
Then, \( A = 882\times0.9659\approx882\times0.9659 \)
Calculate \( 882\times0.9659 \approx 851.92 \)

Step3: Round to the nearest square foot

Rounding \( 851.92 \) to the nearest whole number gives \( 852 \).

Answer:

\( 852 \)