3.5k views
0 votes
Please help precalc optimization problem​

Please help precalc optimization problem​-example-1
User Joe Todd
by
8.2k points

1 Answer

4 votes

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

User Petezurich
by
8.3k points

Related questions

asked Oct 21, 2024 148k views
Artparks asked Oct 21, 2024
by Artparks
7.8k points
2 answers
3 votes
148k views
asked Apr 4, 2024 90.2k views
Large asked Apr 4, 2024
by Large
7.6k points
1 answer
0 votes
90.2k views
asked Apr 14, 2024 30.0k views
Ed Pavlov asked Apr 14, 2024
by Ed Pavlov
7.9k points
1 answer
3 votes
30.0k views