A rancher wants to fence in an area of 500000 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?
3464.1016
I will set it up for you.
Let the length of the rectangle be y feet
let its width be x feet. The divider will then be x feet
So the length L
= 3x + 2y
but you know that xy = 500000
and y = 500000/x
then
L = 3x + 2(500000/x)
differentiate, set that equal to zero and solve for x
Put x back into L = ...
and you are done
To find the shortest length of fence that the rancher can use, we need to determine the dimensions of the rectangular field first. Let's assume the length of the field is L and the width is W.
Since the total area of the field is given as 500000 square feet, we can write the equation:
L x W = 500000
As the rancher wants to divide the field into two equal halves with a fence down the middle, one half of the field will have an area of 500000 / 2 = 250000 square feet.
Since the fence runs down the middle, it divides the field into two equal width sections. So, the width of each section, represented as W/2, remains the same. The length of each section is L.
For one section, the equation for area becomes:
L x (W/2) = 250000
By rearranging this equation, we can solve for L:
L = 250000 / (W/2)
L = 500000 / W
Now, we have two equations:
L x W = 500000
L = 500000 / W
We can substitute the value of L in the first equation with 500000 / W:
(500000 / W) x W = 500000
Simplifying this equation, we can cancel out W and find the value of W:
500000 = W^2
Now, we can solve for W by taking the square root of both sides:
W = √500000
W ≈ 707.1
Since the rancher wants to find the shortest length of fence, we need to find the perimeter of the divided field. The perimeter is given by:
P = 2L + W
Substituting the values of L and W:
P = 2(500000 / W) + W
P = 2(500000 / 707.1) + 707.1
P ≈ 2114.2
Therefore, the shortest length of fence that the rancher can use is approximately 2114.2 feet.