A veterinarian has 100 feet of fencing and wishes to construct 6 dog kennels by first building a fence around a rectangular region, and then subdividing that region into six smaller regions. What dimensions will maximize the total area?
Let x be the length of the rectangular region and y be the width of the rectangular region.
The total area of the rectangular region is A = xy.
The total amount of fencing is 100 feet.
Therefore, 2x + 2y = 100
Substituting this into the equation for A, we get A = x(100 - 2x)/2.
Taking the derivative of A with respect to x, we get A' = (100 - 4x)/2.
Setting A' = 0 and solving for x, we get x = 25.
Substituting this into the equation for 2x + 2y = 100, we get y = 25.
Therefore, the dimensions that will maximize the total area are x = 25 and y = 25.