A landscaper is designing a flower garden in the shape of a trapezoid. She wants the shorter base to be 3 yards greater than the height, and the longer base to be 7 yards greater than the height. She wants the area to be 225 square yards. The situation is modeled by the equation, one-half h multiplied by b one plus b two, where h is the height, in yards, and b1 and b2 are the length of the two bases, in yards. Complete the square to find the height that will give the desired area. Round to the nearest hundredth of a yard.
To find the height that will give the desired area for the flower garden, we need to complete the square using the given equation.
The equation for the area of a trapezoid is:
Area = (1/2) * Height * (Base1 + Base2)
In this case, the shorter base is 3 yards greater than the height, so we can express it as:
Base1 = Height + 3
The longer base is 7 yards greater than the height, so we can express it as:
Base2 = Height + 7
Substituting these values into the area equation, we get:
225 = (1/2) * Height * (Height + 3 + Height + 7)
Simplifying further:
225 = (1/2) * Height * (2 * Height + 10)
Dividing both sides by 1/2:
450 = Height * (2 * Height + 10)
Expanding the equation:
450 = 2 * Height^2 + 10 * Height
Rearranging the equation to have the quadratic term on the left side:
2 * Height^2 + 10 * Height - 450 = 0
Now we can complete the square to solve for the height:
Divide the entire equation by 2 to simplify:
Height^2 + 5 * Height - 225 = 0
To complete the square, we need to add the square of half the coefficient of the linear term (5/2) to both sides of the equation:
Height^2 + 5 * Height + (5/2)^2 - (5/2)^2 - 225 = 0
Simplifying further:
Height^2 + 5 * Height + 25/4 - 25/4 - 225 = 0
Combining like terms:
Height^2 + 5 * Height + 25/4 - 901/4 = 0
Height^2 + 5 * Height - 876/4 = 0
Height^2 + 5 * Height - 219 = 0
Now, we can use the quadratic formula to solve for the height:
Height = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 5, and c = -219:
Height = (-5 ± sqrt(5^2 - 4 * 1 * -219)) / (2 * 1)
Height = (-5 ± sqrt(25 + 876)) / 2
Height = (-5 ± sqrt(901)) / 2
Since we are only interested in positive values of height, we can ignore the negative solution:
Height = (-5 + sqrt(901)) / 2
Using a calculator, we can find the approximate value:
Height ≈ 9.82 yards
Rounding to the nearest hundredth of a yard, the height that will give the desired area is approximately 9.82 yards.