Question

a painter is going to apply a special coating to a triangular metal plate on a new building. two sides measure 11.3 cm and 12.6 cm. she knows that the angle between the two sides is 110°. what is the area of the surface she plans to cover with the coating?
the surface area is □ cm².
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

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 \( \theta \) are known is \( A=\frac{1}{2}ab\sin\theta \). Here, \( a = 11.3 \) cm, \( b = 12.6 \) cm, and \( \theta=110^\circ \).

Step2: Substitute the values into the formula

First, calculate \( \sin(110^\circ) \). Using a calculator, \( \sin(110^\circ)\approx\sin(180^\circ - 70^\circ)=\sin(70^\circ)\approx0.9397 \). Then, \( A=\frac{1}{2}\times11.3\times12.6\times\sin(110^\circ) \). Calculate \( \frac{1}{2}\times11.3\times12.6 = \frac{11.3\times12.6}{2}=\frac{142.38}{2} = 71.19 \). Then multiply by \( \sin(110^\circ) \): \( 71.19\times0.9397\approx71.19\times0.94\approx66.92 \) (more accurately, \( 71.19\times0.9397\approx66.9 \) when rounded to the nearest tenth).

Answer:

\( 66.9 \)