A farmer has 520 feet of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is 310 feet. Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions. (Hint: Be mindful of the domain of the function you are maximizing.)

Step-by-step explanation:

GIVEN: A farmer has of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is .

TO FIND: Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions.

SOLUTION:

Let the length of rectangle be and

perimeter of rectangular pen

area of rectangular pen

putting value of

to maximize

but the dimensions must be lesser or equal to than that of barn.

therefore maximum length rectangular pen

                              width of rectangular pen

Maximum area of rectangular pen

Hence maximum area of rectangular pen is and dimensions are