First we can find the length of 'a' using law of cosines:
[tex]\begin{gathered} a^2=b^2+c^2-2bc\cdot\cos (A) \\ a^2=32^2+19^2-2\cdot32\cdot19\cdot\cos (47\degree) \\ a^2=1024+361-1216\cdot0.682 \\ a^2=1385-829.31 \\ a^2=555.69 \\ a=23.57 \end{gathered}[/tex]
Now we can calculate the area of the triangle using Heron's formula:
[tex]Area=\sqrt[]{p(p-a)(p-b)(p-c)}[/tex]
Where p is the semiperimeter of the triangle. So we have that:
[tex]\begin{gathered} p=\frac{32+19+23.57}{2} \\ p=37.285 \\ \text{Area}=\sqrt[]{37.285(13.715)\mleft(5.285\mright)\mleft(18.285\mright)} \\ \text{Area}=222.3 \end{gathered}[/tex]
So the area of the triangle is 222.3 ft².
