suppose a triangle has an area of 10ft^2. IF the base of the triangle is 1 foot longer than the height of the triangle, what are the lenghts of the base and height?
Please help and show work
1/2 * b * h = 10
b = h + 1
substituting ... h^2 + h - 20 = 0
(h + 5)(h - 4) = 0
h - 4 = 0
To solve this problem, we need to set up an equation using the formula for the area of a triangle: Area = (1/2) * base * height.
Let's say the height of the triangle is x feet. According to the problem, the base is 1 foot longer than the height, so the base can be represented as (x + 1) feet.
Now we can substitute these values into the equation and solve for x:
10 = (1/2) * (x + 1) * x
To simplify, we can remove the fraction by multiplying both sides of the equation by 2:
20 = (x + 1) * x
Expanding the equation, we get:
20 = x^2 + x
Rearranging the equation to the standard quadratic form, we have:
x^2 + x - 20 = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Since this equation can be factored easily, let's use that method to find the values of x:
(x + 5)(x - 4) = 0
This quadratic equation has two possible solutions: x = -5 or x = 4.
Since we are dealing with measurements, a negative value for the height doesn't make sense. Therefore, we discard -5 as a viable solution.
Therefore, the height of the triangle is x = 4 feet. Using this value, we can find the length of the base by adding 1:
Base = height + 1 = 4 + 1 = 5 feet.
So, the length of the base is 5 feet, and the height is 4 feet.