Question 886052
let x represents the length of the side perpendicular to the river.
let y represent the length of the side parallel to the river.
the perimeter is equal to 2x + y
since the cost per foot of x is equal to 8, then the cost of x feet is equal to 8 * x.
since the cost per foot of y is equal to 5, then the cost of y feet is equal to 5 * y.
the cost of the perimeter of fencing is equal to (4 * 25) + (2 * 8 * x) + (5 * y)
the 4 * 25 is the cost of the 4 posts.
the 2 * 8 * x is the cost of the sides perpendicular to the river.
the 5 * y is the cost of the side parallel to the river.


since the area is equal to length * width and x is the width and y is the length, then the area is equal x * y
since the area is equal to 1000 square feet, then you get:
x * y = 1000
if you solve for y, you get y = 1000 / x.


substitute for y in the cost equation and you get:
c(x) = (4 * 25) + (2 * 8 * x) + (5 * (1000 / x))


simplify this equation to get:
c(x) = 100 + (16 * x) + (5000 / x).


the value of x cannot be less than or equal to 0.
the value of x has to be less than infinity.
the value of y will be based on the equation of y = 1000 / x, because the area will always have to be 1000 and the equation for the area is x * y = 1000 and when you solve for y in the equation of x * y = 1000, you get y = 1000 / x.



if you graph this equation, you will find that the minimum cost for the perimeter is when x = 17.677671.
When x = 17.677671, y = 1000 / 17.677671 = 56.56853779
finding this value is not very easy.
I used the TI-84 plus graphing calculator to find it for me.


when x = 17.677671 and y = 56.56853779, the cost for the fencing becomes 615.6854249.


you can see from the graph shown below, that this is the minimum cost for the perimeter.


the domain of the function of c(x) = 100 + (16 * x) + (5000 / x) is equal to all values of x greater than 0.


the range of the function of c(x) is equal to all values of y greater than or equal to 615.6854249.


you can see both of those from the graph as well.




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