a triangle has an area of 35 square inches. The base of the triangle is 4 inches less than twice the height. What is the length of the base of the triangle?
I THINK ITS THIS :
(1/2) x (2h-4) x h = 35
(h-2)h = 35
h^2-2h-35 = 0
(h-7)(h+5) = 0
Positive solution:
height = 7 inches
base = 2h-4 = 2*7-4 = 10 inches
let the height be x
then base = 2x-4
(1/2)(x)(2x-4) = 35
x(2x-4) = 70
2x^2 - 4x - 70 = 0
x^2 - 2x - 35 = 0
(x-7)(x+5) = 0
x = 7 or x = a negative, which is silly
base = 2x-4 = 10
To find the length of the base of the triangle, we'll use the formula for the area of a triangle: A = (1/2) * base * height.
Given that the area of the triangle is 35 square inches, we can substitute the values into the formula: 35 = (1/2) * base * height.
We also know that the base of the triangle is 4 inches less than twice the height. So, we can define the base in terms of the height as follows: base = 2 * height - 4.
Substituting this expression for the base into the area formula, we have: 35 = (1/2) * (2 * height - 4) * height.
Next, let's simplify the equation by distributing and combining like terms: 35 = (height - 2) * height.
Expanding the equation further: 35 = height^2 - 2 * height.
Now, let's rearrange the equation into a quadratic form: height^2 - 2 * height - 35 = 0.
To solve for the height, we can either factor this quadratic equation or use the quadratic formula. Factoring isn't possible in this case, so let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by the formula: x = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 1, b = -2, and c = -35. Substituting these values into the quadratic formula: height = (-(-2) ± √((-2)^2 - 4 * 1 * -35)) / (2 * 1).
Simplifying further: height = (2 ± √(4 + 140)) / 2.
height = (2 ± √144) / 2.
height = (2 ± 12) / 2.
There are two possible solutions for the height: height = (2 + 12) / 2 = 14 / 2 = 7 or height = (2 - 12) / 2 = -10 / 2 = -5.
Since the height of a triangle cannot be negative, we discard the negative solution and choose the positive height, which is 7 inches.
Now we can find the base of the triangle using the equation: base = 2 * height - 4 = 2 * 7 - 4 = 14 - 4 = 10.
Therefore, the length of the base of the triangle is 10 inches.