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:
[tex](76-x)(70-x)=720[/tex]
Operating:
[tex]\begin{gathered} 5320-76x-70x+x^2=720 \\ \text{Simplify:} \\ x^2-146x+5320-720=0 \\ x^2-146x+4600=0 \end{gathered}[/tex]
The coefficients of this quadratic equation are a = 1, b = -146, c = 4600.
To calculate the roots of the equation, use the formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
Substituting:
[tex]\begin{gathered} x=\frac{146\pm\sqrt[]{(-146)^2-4\cdot1\cdot4600}}{2\cdot1} \\ x=\frac{146\pm\sqrt[]{21316-18400}}{2} \\ x=\frac{146\pm54}{2} \end{gathered}[/tex]
We have two possible solutions:
[tex]\begin{gathered} x=\frac{146+54}{2}=\frac{200}{2}=100 \\ x=\frac{146-54}{2}=\frac{92}{2}=46 \end{gathered}[/tex]
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:
[tex]\begin{gathered} A=46^2 \\ A=2116 \end{gathered}[/tex]
The area of the park is 2116 square feet
