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 = ((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 = ((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=(1)/(2) (x+y)z`

`A=225\ yd^(2)`

so

`225=(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=(1)/(2) ((z+3)+(z+7))z`

`225=(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`