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, , 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 of the trapezoid, let's re-arrange the equation to be in the form of (h + k)² = c, where h is the variable, k is a constant, and c is a constant.
The equation of the trapezoid's area is h * ((b1 + b2) / 2) = 225.
First, let's find the average of the two bases: b_avg = (b1 + b2) / 2 = (h + (h + 3) + h + (h + 7)) / 2 = (4h + 10) / 2 = 2h + 5.
We can then plug this average into the equation: h * (2h + 5) = 225.
Expanding this equation gives us: 2h² + 5h = 225.
To complete the square, we'll start by adding (5/2)² = 6.25 to both sides of the equation: 2h² + 5h + 6.25 = 225 + 6.25.
Simplifying both sides of the equation gives us: 2(h + (5/4))² = 231.25.
Dividing both sides of the equation by 2 gives us: (h + (5/4))² = 115.625.
Next, we'll take the square root of both sides of the equation to solve for h + (5/4): sqrt((h + (5/4))²) = ±sqrt(115.625).
Taking the square root gives us: h + (5/4) = ±10.74.
Subtracting (5/4) from both sides of the equation: h = -5/4 ± 10.74.
Therefore, the possible heights of the trapezoid are approximately h = 10.74 - (5/4) and h = -10.74 - (5/4).
Rounding to the nearest hundredth, the heights are approximately h = 2.06 yards and h = -15.06 yards.
Since the height of the trapezoid cannot be negative, the height that will give the desired area is approximately 2.06 yards (rounded to the nearest hundredth).