Question 886052: Minimum Cost: A rectangular area adjacent to a river is to be fenced in; no fence is needed on the river side. The enclosed area is to be 1,000 square feet. Fencing for the side parallel to the river is $5 per linear foot, and fencing for the other two sides is $8 per linear foot; the four corner posts are $25 apiece. Let x be the length of one of the sides perpendicular to the river.
a) Write a function C(x) that describes the cost of the project.
b) What is the domain of C?
I don't know exactly how to do this. Any suggestions on what I mess up on is greatly appreciated.
4 * 25 = 100 NOT A VARIABLE COST
5y * 8x = 1,000 I put it like this because it is a rectangle and I was trying to get the area. So I multiplied l * w.
5y * 8x - 5y = 1,000 - 5y
8x = -5y + 1,000
8x - 1,000 = -5y + 1,000 - 1,000
8x - 1,000 = -5y
8x - 1,000 / -5 = -5y / -5
-8/5x - 200
C(x) = - 8/5x - 200
I was told it was wrong. I've also tried the following.
8x^2 + 5 y + 100 = 1000
8x^2 5y + 100 - 100 = 1000 - 100
8x^2 + 5y = 900
8x^2 + 5y - 8x^2 = -8x^2 + 900
5y = -8x^2 + 900
5y / 5 = -8x2 + 900 / 5
y = -8/5x^2 +180
8x + 8x + 5y + 100 = 1000
16x + 5y + 100 = 1000
16x + 5y + 100 - 100 = 1000 - 100
16x + 5y = 900
16x + 5y - 16x = -16x + 900
5y = -16x + 900
5y / 5 = -16x + 900 / 5
y = -16/5x + 180
I'm not sure how to explain the cost of the project or to find the domain either.
Thank you in advance. Chastity Moore
Found 2 solutions by richwmiller, Theo:
Answer by richwmiller(17219)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! 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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