Question

a rectangular parking lot has a perimeter of 820 ft. the area of the parking lot measures 42,000 ft². what is a dimension of the parking lot?
120 ft
205 ft
210 ft
375 ft

Explanation:

Step1: Recall the formulas for perimeter and area of a rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\) (where \(P\) is perimeter, \(l\) is length, \(w\) is width), and the area formula is \(A=l\times w\). Given \(P = 820\) ft, so \(l + w=\frac{820}{2}=410\), then \(l = 410 - w\). Substitute into the area formula: \(A=(410 - w)\times w=42000\).

Step2: Solve the quadratic equation

Expand the equation: \(410w-w^{2}=42000\), rewrite it as \(w^{2}-410w + 42000=0\). Use the quadratic formula \(w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here \(a = 1\), \(b=-410\), \(c = 42000\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(-410)^{2}-4\times1\times42000=168100 - 168000 = 100\). Then \(w=\frac{410\pm\sqrt{100}}{2}=\frac{410\pm10}{2}\).

Step3: Find the values of \(w\) and \(l\)

Case 1: \(w=\frac{410 + 10}{2}=210\) ft, then \(l=410 - 210=200\) ft. Case 2: \(w=\frac{410-10}{2}=200\) ft, then \(l=410 - 200=210\) ft.

Answer:

210 ft and 200 ft. So among the options, 210 ft is a dimension of the parking lot.