the altitude of a triangle is 3m less than half its base. If the area of the triangle is 300 m^2 find the measure of the base.
Area = 1/2 Height * Base
If base = x, then height = .5x-3
300 = 1/2x(.5x-3)
Solve for x.
300=1/2x(5x-3)
5x^2-3x-600=0
(using Quadratic formula)
x1=-10.659
x2=11.259
...we have no negative measure for the BASE so,
b = 11.3m
To find the measure of the base of the triangle, we can use the formula for the area of a triangle. The formula is:
Area = (1/2) * base * altitude
Let's start by assigning variables to the unknown quantities. Let's call the base of the triangle "b" and the altitude "h".
According to the given information, the altitude is 3m less than half its base. So we can write the following equation:
h = (1/2)b - 3
Next, we know that the area of the triangle is 300 m^2. So we can substitute the values into the area formula:
300 = (1/2) * b * h
Now, substitute the expression for h from the first equation into the area formula:
300 = (1/2) * b * ((1/2)b - 3)
Simplify the equation:
300 = (1/4)b^2 - (3/2)b
To solve this equation for b, let's multiply both sides by 4 to get rid of the fraction:
1200 = b^2 - 6b
Next, rearrange the equation to a quadratic equation form:
b^2 - 6b - 1200 = 0
Now, we will solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, we can factorize the quadratic equation:
(b - 50)(b + 24) = 0
Setting each factor to zero:
b - 50 = 0 or b + 24 = 0
Solving each equation for b:
b = 50 or b = -24
Since we are talking about the length of a base, which cannot be negative, we discard the negative value. Therefore, the base of the triangle is 50 meters.