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 . Use the Quadratic Formula to find the height that will give the desired area. Round to the nearest hundredth of a yard.

The equation is to be given, judging by the tone problem given but apparently it's not given. The equation can be derived from the problem. The area of a trapezoid is given by the equation \(A = \frac{(a+b)}{2} h\). Also, given that a, is the longer side, then a = h+7 and b (shorter side) = h+3 and h is the height. the equation is then \(A = \frac{(h+3)(h+7)}{2}h = 225\). Solving this quadratic equation, h is 4.84.

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calculista calculista · Answer 2 · 2018-10-03T09:49:41-04:00

Let

x--------> shorter base in yards

y--------> the longer base in yards

z-------> the height in yards

we know that

the area of the flower garden in the shape of a tra-pezoid is

\(A=\frac{1}{2} (x+y)z\)

\(A=225\ yd^{2}\)

so

\(225=\frac{1}{2} (x+y)z\) --------> equation \(1\)

\(x=z+3\) --------> equation \(2\)

\(y=z+7\) --------> equation \(3\)

Substitute equation \(2\) and equation \(3\) in equation \(1\)

\(225=\frac{1}{2} ((z+3)+(z+7))z\)

\(225=\frac{1}{2} ((z+3)+(z+7))z\\ \\450=[2z+10]z\\ \\2z^{2}+10z-450=0\)

Solve the quadratic equation for z

using a graphing tool

see the attached figure

the solution is

\(z=12.71\ yd\)

therefore

the answer is

the height of the flower garden in the shape of a tra-pezoid is \(12.71\ yd\)