Question

As a property owner, you want to fence a garden which is adjacent to a road. The fencing next to the road must be stronger and cost $6 per foot. The fencing on the other sides cost $4 per foot. The area of garden is 2400 square feet.

1. Draw several diagrams to express the situation and calculate the cost for each configuration, and then estimate the dimension of minimum cost.

2. Find the function that represents the cost in terms of one of its sides.

3. Using the graphing utility, graph this cost function

4. Find the dimension that minimizes the cost of fencing and compare this with your estimate based the diagram you made.

Answer 1

Diagram:

The area of garden is 2400 square feet

x * y = 2400

y = 2400/x

The cost of  fencing thee side x of this garden is $6 per foot and the rest three sides can be fenced for $4 per foot

So the total cost of fencing this garden:

C=6x + 4(x+2y)

C = 6x + 4x + 8y

C = 10x + 8y

Since y = 2400/x

Graphing the cost function:

minimizing the cost of fencing:

Lets find the derivative of C(x)

Now put C'(x) =0,we get

Find the double derivative

Put both values of x in C''(x) and the minimum will occur when C''(x) >0

So the minimum cost occurs when and

y = 2400/x