janet worked on a garden shaped like a triangle with a base that is 5 m less than twice the height. If the area of the garden is 90 m^2, how long is the base and the height?
90 = 1/2 (2h-5) h
h = (5+√1465)/4
b = 2h-5 = 1/2 (√1465 - 5)
odd answer, so better check my math
To find the length of the base and height of the garden, we can start by setting up variables for the base and height.
Let's represent the height of the triangle as 'h' meters.
According to the problem, the base of the triangle is 5 meters less than twice the height, so the length of the base can be represented as (2h - 5) meters.
Now, we have the formula for the area of a triangle: Area = (base * height) / 2.
Plugging in the given area (90 m^2) along with the variables we defined, we can set up the equation as follows:
90 = ((2h - 5) * h) / 2
To solve this equation, we can eliminate the division by multiplying both sides of the equation by 2:
2 * 90 = (2h - 5) * h
180 = 2h^2 - 5h
Rearranging the equation:
2h^2 - 5h - 180 = 0
Now, we have a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula.
Factoring and solving this equation, we find:
(2h + 15)(h - 12) = 0
This equation has two possible solutions:
1) 2h + 15 = 0
=> 2h = -15
=> h = -15/2 (Discard this solution since height cannot be negative)
2) h - 12 = 0
=> h = 12
So, the height of the garden is 12 meters.
To find the length of the base, we substitute the value of 'h' back into our expression for the base:
Base = 2h - 5
= 2 * 12 - 5
= 24 - 5
= 19
Therefore, the length of the base is 19 meters.