Answer:
225/4
Explanation:
If x is one leg, and 15 is the hypotenuse, then the other leg (let's call it y) can be found with Pythagorean theorem:
15² = x² + y²
y² = 225 − x²
y = √(225 − x²)
The area of the triangle is:
A = ½ xy
A = ½ x √(225 − x²)
To find the maximum of A, find dA/dx and set it equal to 0. As suggested by the problem, it may help to square both sides first:
A² = ¼ x² (225 − x²)
A² = ¼ (225x² − x⁴)
Take derivative using power rule and chain rule:
2A dA/dx = ¼ (450x − 4x³)
0 = ¼ (450x − 4x³)
0 = 450x − 4x³
0 = 2x (225 − 2x²)
x = 0 or 15/√2.
x = 0 is a minimum. x = 15/√2 is the maximum. So the maximum area is when the triangle is a 45-45-90 triangle.
A = ½ xy
A = ½ (15/√2) (15/√2)
A = 225/4