Question

a painter is going to apply a special coating to a triangular metal plate on a new building. two sides measure 13.4 cm and 15.6 cm. she knows that the angle between the two sides is 108°. 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, \( a = 13.4 \) cm, \( b = 15.6 \) cm, and \( \theta=108^\circ \).

Step2: Substitute the values into the formula

First, calculate \( ab \): \( 13.4\times15.6 = 209.04 \). Then, calculate \( \sin(108^\circ) \). We know that \( \sin(108^\circ)=\sin(180^\circ - 72^\circ)=\sin(72^\circ)\approx0.9511 \). Now, substitute into the area formula: \( A=\frac{1}{2}\times209.04\times0.9511 \).

Step3: Perform the calculation

First, \( \frac{1}{2}\times209.04 = 104.52 \). Then, \( 104.52\times0.9511\approx104.52\times0.9511 \approx 99.4 \) (rounded to the nearest tenth).

Answer:

\( 99.4 \)