A rancher wants to fence in an area of 1500000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

Area to be fenced in, A = 1,500,000 sq.ft.

Width (shorter side) = x
Length (longer side) = A/x

Total length of fence, L
= 3x+2(A/x)

Differentiate with respect to x:
dL/dx = 3 - 2A/x²

To get minimum length,
dL/dx = 0
3 - 2A/x² = 0
x=sqrt(2A/3)=1000 ft.

Check that d²L/dx²>0 for L to be a minimum.

Calculate the length of fence required.

thank you so much for the clear explanation. i really understand how to do the problem now! i appreciate it!

You're welcome!

what is the answer of this question?

To find the shortest length of fence, we need to determine the dimensions of the rectangular field first. Let's assume the length of the field is L feet and the width is W feet.

The total area of the rectangular field is given as 1,500,000 square feet. Therefore, we can write the equation for the area as:

L * W = 1,500,000

We are also told that the field needs to be divided in half with a fence down the middle parallel to one side. This means that one part of the field will have a length of L/2 feet and the other part will also have a length of L/2 feet.

Now, let's solve for the dimensions of the field. We can rewrite the equation for area in terms of L:

L * W = 1,500,000
L = 1,500,000 / W

Substituting L/2 for L in the equation, we get:

(L/2) * W = 1,500,000
L * W = 3,000,000

Comparing this equation with the original equation for the area, we can see that the area of the field when it is divided in half is twice the original area. Therefore, the dimensions of the field when divided in half are:

L = √(3,000,000)
W = 1,500,000 / √(3,000,000)

Now that we know the dimensions of the field, we can calculate the shortest length of fence needed. There are four sides of the rectangular field, and two sides need to be divided by the fence. So, the length of the fence required in this case will be:

Length of fence = 2L + W

Substituting the values of L and W, we get:

Length of fence = 2 * √(3,000,000) + (1,500,000 / √(3,000,000))

Calculating this equation will give us the shortest length of fence that the rancher can use.