the height of a triangle is 10m longer than twice its base. its area is 176m^2. Find the lengths of its height and base.

base = x

height = 2x+10

(1/2)(x)(2x+10) = 176
2x^2 + 10x = 352
x^2 + 5x - 176 = 0
(x + 16)(x-11) = 0
x = -16 or x = 11 , but x > 0

so x = 11

base is 11, height is 32

check: (1/2)(11)(32) = 176

To find the lengths of the height and base of the triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

We are given that the area is 176m^2. Therefore, we can write the equation as:

176 = (1/2) * base * height

We are also given that the height is 10m longer than twice the base. So we can write the height in terms of the base as:

height = 2 * base + 10

Now we can substitute this expression for height into the equation for the area:

176 = (1/2) * base * (2 * base + 10)

Let's simplify this equation and solve for the base:

176 = base * (base + 5)

Expanding the equation:

176 = base^2 + 5base

Rearranging the equation by subtracting 176 from both sides:

base^2 + 5base - 176 = 0

Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:

(base + 16)(base - 11) = 0

This gives us two possible values for the base: base = -16 or base = 11. Since the length of a side cannot be negative, we discard the negative solution.

So the base of the triangle is 11m.

To find the height, we can substitute this value back into the expression for height:

height = 2 * base + 10
= 2 * 11 + 10
= 32

Therefore, the length of the height is 32m and the length of the base is 11m