Afarmer wishes to enclose a rectangular region bordering a barn with fencing. suppose that x represents the length of each of the three parallel pieces of fencing. she has 600 feet of fencing available. a. what is the length of the remaining pieces of fencing in terms of x? b. determine a function that represents the total area of the enclosed region. give the restrictions on x. c. what dimensions for the total enclosed region would give an area of 22,500 square feet? also, how do i find the maximum area enclosed?

a)

The 3 is because we have 3 parallel pieces.

And then if we solve for h we got:

b)

And if we analyze the function we see that since we can't have a negative area.

c)  

And in order to find the maximum area enclosed we need to derivate respect h, the original function.

And as we can see then v= 100 and h = 600-3(100) = 300

And the maximum area would be given by :

Step-by-step explanation:

For this case we can assume the following notation:

represent the vertical dimension

represent the horizontal dimension

We know also that the total length of fencing is 600

Part a

So then we can create the following function:

The 3 is because we have 3 parallel pieces.

And then if we solve for h we got:

Part b

For this case since we have a rectangular area we have the following formula for the area:

And if we analyze the function we see that since we can't have a negative area.

Part c

For this case we want to find the dimensions that maximize a total area of 22500, so we have this expression:

We cna express this function like this:

And as we can see we have a quadratic equation and we can solve it with the following formula:

For this case

And if we replace we got:

If we use 150 for v then the value of h would be 0 and that not makes sense so then the best solution would be v = 50

And then we can solve for h like this:

And in order to find the maximum area enclosed we need to derivate respect h , the original function.

And as we can see then v= 100 and h = 600-3(100) = 300

And the maximum area would be given by :