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.
(1 point)
Responses
25.41 yards
25.41 yards
460 yards
460 yards
15.21 yards
15.21 yards
12.71 yards
The equation for the area of the trapezoid is given as:
A = (1/2)h(b1 + b2)
We are given that the shorter base (b1) is 3 yards greater than the height (h), and the longer base (b2) is 7 yards greater than the height.
From the given information, we can write the equation:
225 = (1/2)h((h + 3) + (h + 7))
Simplifying this equation, we get:
225 = (1/2)h(2h + 10)
Distributing the (1/2)h, we have:
225 = h^2 + 5h
Rearranging the equation in the standard form of a quadratic equation, we have:
h^2 + 5h - 225 = 0
Completing the square, we add and subtract the square of half the coefficient of the h term (5/2)^2 = 6.25 to the equation:
h^2 + 5h + 6.25 - 6.25 - 225 = 0
(h + 2.5)^2 - 6.25 - 225 = 0
(h + 2.5)^2 = 231.25
Taking the square root of both sides, we get:
h + 2.5 = ±√231.25
h = -2.5 ± √231.25
Since the height cannot be negative, we take the positive square root:
h = -2.5 + 15.21 = 12.71
So, the height that will give the desired area of 225 square yards is approximately 12.71 yards.