Question

A real estate company wants to build a parking lot along the side of one of its buildings using 400 feet of fence. If the side along the building needs no fence, what are the dimensions of the largest possible parking lot?
1. side parallel to the building ______ ft 2. side perpendicular to the building _____ ft

Answer 1

Answer:

1.) Therefore the side parallel to the building is 100 feet.

2.) The side perpendicular to the building is 200 feet.

Step-by-step explanation:

The parking lot, which is rectangular,  is built along the side of one of its buildings.

The side of the parking lot which is alongside the building needs no fence,

let us say that this side is of length y.

Let the perpendicular to the building be of length x.

The perimeter of the parking lot that actually needs to be fenced  = 2x + y

The amount of fencing available  = 400 feet.

Therefore 2x + y = 400    ....    (i)

Therefore y = 400 - 2x     ....    (ii)

The area of the parking lot, A = xy   =   x ( 400 - 2x)  =   400x - $2x^2$   ....   (iii)

Therefore $\frac{dA}{dx} = \frac{d(400x - 2x^2}{dx} = 400 - 4x$     ....   (iv)

Also we get $\frac{d^2A}{dx^2} = -4$     ....    (v)

Hence we will get the maximum value of the parking lot by equating the first order differential of area, A to zero as the second order differential of area, A is negative.

Therefore from (iv) we get 400 - 4x  = 0     ∴ x = 100 feet

y  = 400 - 2x = 400 - 200 = 200 feet.

1.) Therefore the side parallel to the building is 100 feet.

2.) The side perpendicular to the building is 200 feet.