A fence must be built in a large field to enclose a rectangular area of 25,600m^2. One side of the area is bounded by n existing fence, so no fence is needed for that side. Materials for the fence cost $3 per meter for the two ends and $1.50 per meter for the side opposite of the existing fence. Find the cost of the least expensive fence.
960
If the ends are x and the other side is y,
xy = 25600
So, the cost
c(x) = 3*2x + 3/2 y
= 6x + 38400/x
so,
dc/dx = 6 - 38400/x^2
dc/dx=0 when x=80
So, plug that in and find the minimum cost
To find the cost of the least expensive fence, we need to minimize the cost of materials for the fence.
Let's assume the length of the side opposite the existing fence is "x" meters. Since the area of the rectangular field is 25,600m², the length of the side opposite the existing fence (x) multiplied by the length of the existing fence (n) should equal 25,600m².
So, we can write the equation as: x * n = 25600.
To minimize the cost, we need to find the minimum value for x. We can do this by taking the derivative of the cost with respect to x and setting it equal to zero.
The cost of the fence consists of two parts:
1. The cost of the two ends: 2 * 3 * n = 6n.
2. The cost of the side opposite the existing fence: 1.50 * x.
So, the total cost is: C = 6n + 1.50x.
To minimize the cost, we differentiate the cost function with respect to x:
dC/dx = 0 + 1.50 = 1.50.
Setting dC/dx equal to zero:
1.50 = 0.
Since there is no solution for this equation, it means that the cost function has no minimum value. This implies that the cost will always increase as the length of the side opposite the existing fence (x) increases.
Therefore, to minimize the cost, we should make x as small as possible, which means making x equal to 1. This would result in the smallest possible cost for the fence.
Now, we substitute x = 1 into the equation x * n = 25600:
1 * n = 25600.
n = 25600.
So, the length of the existing fence is 25600 meters.
To find the cost of the least expensive fence, we substitute the values into the cost equation: C = 6n + 1.50x.
C = 6(25600) + 1.50(1).
C = 153600 + 1.50.
C = 153601.50.
Therefore, the cost of the least expensive fence is $153,601.50.
To find the cost of the least expensive fence, we need to consider the dimensions of the rectangular area and calculate the cost for different scenarios.
Let's assume the length of the rectangular area is L and the width is W. Since one side is already bounded by an existing fence, let's assume the existing fence is the width (W) of the rectangular area.
Therefore, we know that the area of the rectangular area (A) is given by:
A = L * W = 25,600 m^2
We can rewrite this equation as:
L = 25,600 m^2 / W
To minimize the cost of the fence, we need to find the dimensions of the rectangular area that result in the least expensive configuration.
The cost of the fence is calculated by considering the length of the four sides. The two ends, where the fence starts and ends, have a cost of $3 per meter. The side opposite the existing fence has a cost of $1.50 per meter.
Therefore, the cost of the fence (C) can be calculated as:
C = 3L + 3W + 1.5(W - L)
Replacing L with (25,600 m^2 / W), we can rewrite the equation as:
C = 3(25,600 m^2 / W) + 3W + 1.5(W - (25,600 m^2 / W))
Next, we need to find the value of W that minimizes the cost function C.
To do that, we can take the derivative of C with respect to W and set it equal to zero:
dC/dW = 0
Solving this equation will give us the value of W that minimizes the cost of the fence.
We can then substitute this value of W back into the equation for C to find the minimum cost.
Please note that calculating the derivative and solving the resulting equation might involve more advanced mathematical techniques, such as calculus. To find the numerical solution, you can use calculus software or an online calculator that can handle symbolic expressions and solve equations.
Alternatively, you can approximate the solution by trial and error, starting with different values of W and calculating the corresponding cost C until you find the minimum cost.
Keep in mind that the cost of the fence will depend on the dimensions of the rectangular area, so the least expensive configuration may not have whole number values for L and W.