Question

A farmer wants to fence an area of 60,000 m2 in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. let y represent the length (in meters) of a side perpendicular to the dividing fence, and let x represent the length (in meters) of a side parallel to the dividing fence. let f represent the length of fencing in meters. write an equation that represents f in terms of the variable x. f(x) = incorrect: your answer is incorrect. find the derivative f ′(x). f ′(x) = incorrect: your answer is incorrect. find the critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.) x = what should the lengths of the sides of the rectangular field be (in m) in order to minimize the cost of the fence? smaller value m larger value m

Answer 1

The equation that represents the length of fencing in terms of the variable x is f(x) = x + x + y + (y/2). The derivative of f(x) is 2. There are no critical numbers for the function f(x).

Step-by-step explanation:

To write an equation that represents the length of fencing in terms of the variable x, we need to consider the perimeter of the rectangular field. The perimeter consists of the sum of all four sides. Since we want to divide the field in half with a fence parallel to one of the sides, the length of that side (which is x) will be divided by 2. Therefore, the equation that represents f, the length of fencing, in terms of x is:

f(x) = x + x + y + (y/2)

To find the derivative f'(x), we differentiate the equation with respect to x:

f'(x) = 1 + 1 + 0 + 0 = 2

To find the critical numbers of the function, we set the derivative equal to zero and solve for x:

2 = 0

There is no solution to this equation. Therefore, there are no critical numbers for the function f(x).