A high school has a rectangular parking lot that measures 600 ft. long by 400 ft. wide. The School Board wants to double the area of the lot by increasing both its length and width by the same amount, x ft.
The Board also wants to build a fence around the new lot. The cost to
expand the lot is estimated to be $2 per square foot of new space.
The cost to fence the lot is
estimated to be $30 per foot of fencing. Costs include materials and labor.
What is the estimated cost of expanding and fencing in the lot?
To double the area of the parking lot, we first need to find the current area and then determine the increase in area needed.
Current area of the parking lot = length * width
Current area = 600 ft * 400 ft = 240,000 sq ft
To double the area, we need to find the new area:
New area = 2 * 240,000 sq ft = 480,000 sq ft
Let's denote the increase in length and width as x ft.
The new length = 600 ft + x ft
The new width = 400 ft + x ft
The new area can also be expressed as:
New area = (600 ft + x ft) * (400 ft + x ft)
Setting this equal to the new area we determined earlier:
(600 ft + x ft) * (400 ft + x ft) = 480,000 sq ft
Expanding this expression:
240,000 + 1000x + x^2 = 480,000
x^2 + 1000x - 240,000 = 0
Using the quadratic formula to solve for x:
x = (-1000 ± sqrt(1000^2 - 4*1*(-240,000)) / (2*1)
x ≈ 316.18 ft
So the increase in length and width is approximately 316.18 ft.
Now, let's calculate the cost to expand the lot:
Cost to expand = $2 * new area
Cost to expand = $2 * 480,000 sq ft
Cost to expand = $960,000
Next, let's calculate the perimeter of the new lot to determine the cost to fence the lot:
Perimeter = 2(600 ft + 316.18 ft) + 2(400 ft + 316.18 ft)
Perimeter = 2(916.18 ft) + 2(716.18 ft)
Perimeter ≈ 3264.72 ft
Cost to fence = $30 * perimeter
Cost to fence = $30 * 3264.72 ft
Cost to fence = $97,941.60
Therefore, the estimated cost of expanding and fencing in the lot is $960,000 (expanding) + $97,941.60 (fencing) = $1,057,941.60.