If the height of a triangle is one inch more than twice its base, and if the area of the same triangle is 20 square inches, find the base and height of the triangle.

Did you forget that area = 1/2 * base * height?

20 = 1/2 * b * (1+2b)
b = 4.229in

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

Area = (base * height) / 2

Since we know that the area is 20 square inches, we can substitute this value into the formula:

20 = (base * height) / 2

Now, let's focus on finding the relationship between the base and height, using the given information. We're told that the height of the triangle is one inch more than twice its base. Let's express this relationship mathematically:

height = 2 * base + 1

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

20 = (base * (2 * base + 1)) / 2

Simplifying this equation, we get:

40 = base * (2 * base + 1)

Expanding the brackets:

40 = 2 * base^2 + base

Rearranging the equation to standard quadratic form:

2 * base^2 + base - 40 = 0

Now, we can solve this quadratic equation to find the value(s) of the base. We can either factorize or use the quadratic formula. Let's use the quadratic formula:

base = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 2, b = 1, and c = -40. Substituting these values into the quadratic formula, we get:

base = (-(1) ± √((1)^2 - 4(2)(-40))) / (2(2))

Simplifying further:

base = (-1 ± √(1 + 320)) / 4

base = (-1 ± √(321)) / 4

Calculating the square root:

base ≈ (-1 ± 17.93) / 4

This gives us two potential solutions for the base:

base ≈ (16.93) / 4 ≈ 4.23

or

base ≈ (-18.93) / 4 ≈ -4.73

Since the base cannot be negative, we discard the second solution. Therefore, the base of the triangle is approximately 4.23 inches.

To find the height, we substitute the value of base into the relationship we established earlier:

height = 2 * base + 1
height ≈ 2 * 4.23 + 1 ≈ 9.46

Therefore, the base of the triangle is approximately 4.23 inches, and the height is approximately 9.46 inches.