a farmer wishes to fence off a rectangular plot of land, using an existing wall as one of the sides . the total are enclosed must be 600 square yards. the fence on the side parallel to the wall will cost 20$ per yard, while the fences on the other side will cost 30$ per yard.
What should the dimension of the rectangle be in order to minimize the total cost of the fence?
let the two equal width be x
let the single side by y
xy = 600
y = 600/x
cost = 20y + 30x
= 20(600/x) + 30x
d(cost)/dx = -12000/x^2 + 30
= 0 for a min of cost
30 = 12000/x^2
x^2 = 400
x = 20
the two equal sides are 20 yds each, and the long side is 600/20 = 30 yds.
To find the dimensions of the rectangle that will minimize the total cost of the fence, we first need to set up an equation for the total cost in terms of the dimensions.
Let's say the width of the rectangle, perpendicular to the existing wall, is x yards. Since the existing wall is one side of the rectangle, the length of the rectangle, parallel to the wall, doesn't need a separate variable. We can express the length in terms of the width as follows:
length = 600 / x (since the total area enclosed is 600 square yards)
Now, let's calculate the cost of the fence:
On the side parallel to the wall:
The cost of the fence on this side is $20 per yard, and the length of this side is the same as the length of the rectangle (i.e., 600 / x yards).
So, the cost of this fence = 20 * (600 / x) = 12000 / x dollars.
On the other two sides:
The cost of the fence on each of these sides is $30 per yard, and the width of the rectangle is x yards.
So, the cost of the fences on these sides = 2 * 30 * x = 60x dollars.
Adding up the costs on all three sides, we get the total cost:
Total cost = Cost of the fence on the side parallel to the wall + Cost of the fences on the other two sides
Total cost = 12000 / x + 60x dollars.
To minimize the total cost, we can differentiate Total cost with respect to x and set it to zero, then solve for x:
d(Total cost) / dx = -12000 / x^2 + 60 = 0
Solving the above equation, we get:
12000 / x^2 = 60
x^2 = 12000 / 60
x^2 = 200
x = √200
x ≈ 14.142
So, the width of the rectangle that will minimize the total cost is approximately 14.142 yards.
To find the length of the rectangle, we can substitute the value of x into the equation for the length:
length = 600 / x
length = 600 / 14.142
length ≈ 42.426 yards.
Therefore, the dimensions of the rectangle that will minimize the total cost of the fence are approximately 14.142 yards (width) and 42.426 yards (length).