SOLUTION: A right triangle has area 84 ft2 and a hypotenuse that is 1 foot longer than one of its sides. (a) If x denotes the length of this side, does 2x^3 + x^2 − 28,224 = 0 ? (b) Is

Question 1146985: A right triangle has area 84 ft2 and a hypotenuse that is 1 foot longer than one of its sides.
(a) If x denotes the length of this side, does 2x^3 + x^2 − 28,224 = 0 ?
(b) Is there a positive root of the equation in part (a) that is less than 28?
(c) Find the lengths of the sides of the triangle.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Since the area is a whole number and the hypotenuse is 1 foot longer than one of the sides, the side lengths are almost certainly a Pythagorean triple.

One of the smallest Pythagorean triples satisfies all the conditions of the problem; 7-24-25.

The area is (1/2)(7)(24) = 84
The hypotenuse is 1 foot longer than one of the sides.

Brute force calculations or a calculator show that x=24 is a root of the equation shown in (a).

ANSWERS:
(a) yes; 24 is a root of that equation
(b) yes; 24 is less than 28
(c) 7, 24, and 25

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