Answer:
The area of the triangle is 2.97 ft².
Step-by-step explanation:
We are given the triangle having sides 2.7 feet, 3.4 feet and the inclusive angle 40°.
We will find the length of the third side.
The law of cosines is given by [tex]c^{2}=a^{2}+b^{2}-2ab\cos \theta[/tex].
Substituting the values, we have,
[tex]c^{2}=2.7^{2}+3.4^{2}-2\times 2.7\times 3.4\cos 40\\\\c^{2}=7.29+11.56-18.36\times 0.766\\\\c^{2}=18.85-14.06\\\\c^2=4.79\\\\c=2.2[/tex]
Thus, the length of the third side is 2.2 feet.
Now, the Heron's Area Formula is given by [tex]A=\sqrt{s(s-a)(s-b)(s-c)}[/tex], where [tex]s=\frac{a+b+c}{2}[/tex].
So, [tex]s=\frac{2.7+3.4+2.2}{2}\\\\s=\frac{8.3}{2}\\\\s=4.15[/tex]
Then, the area of the triangle is,
[tex]A=\sqrt{4.15(4.15-2.7)(4.15-3.4)(4.15-2.2)}\\\\A=\sqrt{4.15\times 1.45\times 0.75\times 1.95}\\\\A=\sqrt{8.80059375}\\\\A=2.97[/tex]
Thus, the area of the triangle is 2.97 ft².
