Question

An event coordinator puts a stage in the northeast comer of asquare-shaped park. The area of the stage is 720 square feet.What is the area of the park?

Answer 1

The park is a square of a side length of x.

The stage in the northeast corner has an area of 720 square feet.

It can be observed the length of the stage is 76 - x feet and the width of the stage is 70 - x feet, thus its area is:

`(76-x)(70-x)=720`

Operating:

`\begin{gathered} 5320-76x-70x+x^2=720 \\ \text{Simplify:} \\ x^2-146x+5320-720=0 \\ x^2-146x+4600=0 \end{gathered}`

The coefficients of this quadratic equation are a = 1, b = -146, c = 4600.

To calculate the roots of the equation, use the formula:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

Substituting:

`\begin{gathered} x=\frac{146\pm\sqrt[]{(-146)^2-4\cdot1\cdot4600}}{2\cdot1} \\ x=\frac{146\pm\sqrt[]{21316-18400}}{2} \\ x=(146\pm54)/(2) \end{gathered}`

We have two possible solutions:

`\begin{gathered} x=(146+54)/(2)=(200)/(2)=100 \\ x=(146-54)/(2)=(92)/(2)=46 \end{gathered}`

Both solutions look like they are valid, but we must recall that the length and the width are 76 - x and 70 - x respectively. If we use x = 100, both dimensions would be negative and it's not acceptable, thus:

x = 46

The area of the park is:

`\begin{gathered} A=46^2 \\ A=2116 \end{gathered}`

The area of the park is 2116 square feet