Question

a farmer has 2400 feet of fence for a ractangular field along a straght river. no fence is neeeded along the reiver. using calculus find the dimensions of the field which has the largest area

Answer 1

This is an Optimization problem.

We know the farmer has 2400 feet of fence, which means that must be the perimeter of the field.

P = 2400

Perimeter is just all the sides added up, since one of the sides is along a river, its just 3 sides or simply

P= 2L + W

We know that Area is just (length x width):

A = (L)(W)

Since you are looking for the maximum AREA, you need to find the highest point on the Area function or the Maximum. You should know that the derivative at Maximums and Minimums are 0, because the tangent lines at the high and low points are just flat. So we just need to find when A' is equal to zero.

First, lets rewrite the Area as a function of one variable, (L). In order to do this, we need to change the (W) to an (L) somehow. Remember the Perimeter? Well that gives us a relationship between L and W.

P= 2L + W P = 2400

2400 = 2L + W

2400 - 2L = W

so plugging the W into the Area function, we get:

A = (L)(2400-2L)

\[A = 2400(L )- (2L^2)\]

Now lets differentiate both sides.

\[A'= 2400 - 4L\]

Since the derivative must be zero at the maximum area, we can just plug in zero:

0 = 2400 - 4L

-2400 = - 4L

L = 600

So the length must be 600. Lets use this to find our Width

W = 2400 - 2L

W = 2400 - 2(600)

W = 2400 - 1200

W = 1200

So the dimensions of the fence must be 600ft x 1200ft which will yield an Area of 480,000 sqft