Arancher wishes to build a fence to enclose a 2250 square yard rectangular field. along one side the fence is to be made of heavy duty material costing $16 per yard, while the remaining three sides are to be made of cheaper material costing $4 per yard. determine the least cost, cmin, of fencing for the rancher.

The least cost of fencing for the rancher is $1200

Step-by-step explanation:

Let x be the width and y the length of the rectangular field.

Let C the total cost of the rectangular field.

The side made of heavy duty material of length of x costs 16 dollars a yard. The three sides not made of heavy duty material cost $4 per yard, their side lengths are x, y, y.  Thus

We know that the total area of rectangular field should be 2250 square yards,

We can say that

Substituting into the total cost of the rectangular field, we get

We have to figure out where the function is increasing and decreasing. Differentiating,

Next, we find the critical points of the derivative

Because the length is always positive the only point we take is . We thus test the intervals and

we see that total cost function is decreasing on and increasing on . Therefore, the minimum is attained at , so the minimal cost is

The least cost of fencing for the rancher is $1200

Here’s the diagram:

the equivalent expression are always the hardest ones but you look at the gcf also

step-by-step explanation: