Farmer Brown wants to fence in a rectangular plot in a large field, using a straight rock wall that is already there as the north boundary. The fencing for the east and west sides of the plot will cost $3 a yard, but she needs to use special fencing, which will minimize the cost of the fencing.

a) find the equation to maximized or minimized.

b) finding the solution.

c)showing that your solution is an absolute maximum or minimum.

how expensive is the special fencing?

To solve this problem, we need to determine the dimensions of the rectangular plot that will minimize the cost of the fencing.

a) To find the equation for minimizing the cost of the fencing, we need to consider the given information. The cost of the east and west sides of the plot is fixed at $3 per yard, but the cost of the north side is already covered by the rock wall. Therefore, the equation we need to minimize is the one that represents the cost of the east and west sides only.

Let's assume the width of the rectangular plot is x yards. Since the north boundary is already defined by the rock wall, the length of the plot would be y yards.

The east and west sides combined have a length of 2y yards. Since the cost per yard is $3, the cost of the east and west sides of the plot would be 2y * $3 = 6y.

Therefore, the equation we need to minimize is: Cost = 6y.

b) To find the solution, we need to rewrite the equation in terms of a single variable. Let's choose x as our variable.

Since the plot is rectangular, the perimeter of the plot would be 2x + 2y. However, we are given that the rock wall defines the north boundary, so the length of the plot (y) is the same as the length of the rock wall. Therefore, the perimeter equation can be simplified to: Perimeter = 2x + y.

We also know that the length of the rock wall (y) is constant. Therefore, to minimize the cost, we need to minimize the value of x.

c) To show that our solution is an absolute minimum, we need to apply the second derivative test. This test helps us determine if our solution indeed results in the lowest cost.

Taking the second derivative of the cost equation, we get: Cost'' = 0. This means that the second derivative is zero, and we have a critical point.

Since the cost equation is linear, it does not contain any concavity. As a result, the second derivative test is inconclusive in this case.

To verify that our solution is the absolute minimum, we can also check the endpoints. If the endpoints give a higher value for the cost equation, our solution would indeed be the absolute minimum. In this case, the endpoints are determined by the boundaries of the field, which are not given.

Therefore, the solution for the dimensions of the rectangular plot that will minimize the cost of the fencing is x = 0, y = constant (same as the length of the rock wall).