Question

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.

Answer 1

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. 

Answer 2

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 $12.71\ yd$