Among all the triangles, the obtuse angle is 150 degrees, and the sum of the sides forming it is 1.6

Among all the triangles, the obtuse angle is 150 degrees, and the sum of the sides forming it is 1.6, find the one with the maximum area. what is the area of this triangle.

The formula for calculating the area of ​​a triangle in terms of an angle and two sides:
S = ½ * x * y * sinα.
α = 150º,
y = 1.6 – x.
Then we rewrite the formula:
S = ½ * x * (1.6 – x) * sin150º.
Sin150º = sin (180º – 30º) = sin180º * cos30º – cos180º * sin30º = 0 – (-1) * (1/2) = ½.
S = ½ * x * (1.6 – x) * sin150º = x * (1.6 – x) / 4, where 0 <X <1.6.
We calculate the maximum of the function x * (1.6 – x) / 4, find its derivative and equate it to zero.
(x * (1.6 – x) / 4) ´ = (1.6 * x / 4 – x ^ 2/4) ´ = (1.6 * x / 4) ´ + (- x ^ 2/4) ´ = 1.6 / 4 – 2 * x / 4 = 0.
Hence,
2 * x = 1.6.
X = 0.8.
At the point x = 0.8, the function takes its maximum value.
Thus, we find the area of ​​a triangle with a maximum area:
S = x * (1.6 – x) / 4 = 0.8 * (1.6 – 0.8) / 4 = 0.8 * 0.8 / 4 = 0.16.