Question

Please help precalc optimization problem​

Answer 1

Answer:

225/4

Step-by-step 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