Answer
$84.40$ m$^{2}$
Work Step by Step
First, we need to find the length of the side $c$ in order to use the formula for the Area of the triangle (SAS):
For this, we first find the angle $B$:
$A^{\circ}+B+C^{\circ}=180^{\circ}$
$59.8^{\circ}+B+53.1^{\circ}=180^{\circ}$
$112.9^{\circ}+B=180^{\circ}$
$B=180^{\circ}-112.9^{\circ}$
$B=67.1^{\circ}$
Now, we use the sine rule to find the length of the side $c$:
$\frac{b}{\sin B}=\frac{c}{\sin C}$
$\frac{15}{\sin 67.1}=\frac{c}{\sin 53.1}$
$c=\frac{15\times\sin 53.1}{\sin 67.1}$
$c=\frac{15\times 0.79968}{0.92119}$
$c=13.021$
The area of the triangle is half the product of the length of two sides and the sine of the angle included between them:
$Area=\frac{1}{2}bc \sin A$
We substitute the values of $A,b$ and $c$ in this formula and solve:
$Area=\frac{1}{2}bc \sin A$
$Area=\frac{1}{2}(15)(13.021) \sin 59.8^{\circ}$
$Area=\frac{1}{2}(195.315) \sin 59.8^{\circ}$
$Area=97.658\sin 59.8^{\circ}$
Using a calculator, $\sin 59.8^{\circ}=0.86427$. Therefore,
$Area=97.658\sin 59.8^{\circ}$
$Area=97.658(0.86427)$
$Area=84.40$
Therefore, the area of the triangle is $84.40$ m$^{2}$.
