The area of a triangle is 384 square feet. Its base is 8 feet longer than its height. Find the length of the base and the height.

Recall that Area of Triangle = (1/2) base height.
Let the height = x. And statement said base = 8 ft longer than height = (x+8)
Therefore: A = (1/2)x(x+8) 384 = x(x+8)/2 Cross Multiplying 768 = x^2 + 8x Re arrange, this is a quadratic equation. x^2 + 8x = 768 x^2 + 8x - 768 = 0. For quadratic factoring we think of two numbers that would multiply to give -768 and add up to give +8.
By search, the numbers are 32 and -24. 32*-24 = -768 and (32+ -24) = 32-24 =8.
Solving: x^2 + 8x - 768 = 0 Replace 8x with (32x -24x) x^2 + 32x -24x -768 =0 x(x+32) -24(x+32) = 0 (x+32)(x -24) = 0 Therefore (x+32) = 0 x = 0-32 =-32. (x-24) = 0 x = 0+24 =24. x = -32 or 24. But our length can not be negative. Therefore x = 24. We throw away the negative answer.
Recall height = x = 24 ft. Base = (x +8) = (24+8) = 32 ft.
Base = 32 ft, Height = 8ft. That's our answer. Hurray!

Answered by olemakpadu | 2024-06-10

The height of the triangle is 24 feet, and the base is 32 feet. The base is 8 feet longer than the height, and both dimensions satisfy the area requirement of 384 square feet. This was calculated using the area formula and a quadratic equation.
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Answered by olemakpadu | 2025-01-13