problem solving

Question

You have 500 feet of fencing which you want to use to enclose a right triangular area. What is the largest area which you can enclose?

Answer 1

you get the maximum area when the right triangle has legs of equal length

let x=each leg

let 500-2x=the hypotenuse

use the Pythagorean theorem to get x²+x²=(500-2x)²

this becomes 2x²=250,000-2000x+4x²

this becomes 2x²-2000x+250,000=0

this becomes x²-1000x+125,000=0

use the quadratic formula to get (1000+√1,000,000-500,000)/2

this becomes (1000+√500,000)/2

(1000+707.1067)/2

292.8932/2=146.4466 feet for each leg of the triangle and 207.1068 feet for the hypotenuse

largest area is...

A=(1/2)bh

A=(1/2)(146.4466)(146.4466)

A=(1/2)21,446.607

A=10,723.303 sq ft