Optimization: Quiz
Problem 1
A rectangular fenced area using 80 feet of fencing is to be created, with one side along an existing wall, so that no fencing needs to be used for that side.
Whare are the dimensions of the largest area that can be created?
Answers, problem 1
The perimeter of the fence is P = 2x + 2y, where x and y are the sides of the rectangle. We are told however that one side of the fencing will be an existing wall, so the perimeter of the fence is actually P = 80 = 2x + y. We also know that the area of the rectangle is A = xy. We can express the area as a function of a single variable, say x. So we can express y in terms of x from the expression for the perimeter of the fencing.
y = 80 - 2x
Therefore A = x(80 - 2x) = 80x - 2x2
dA/dt = 80 - 4x.
The area is a maximum when dA/dt = 0. Therefore 80 - 4x = 0 => x = 20 feet
Plugging the value of x in the equation 80 = 2(20) + y => y = 40 feet
Therefore the dimensions of the largest area that can be created is
20 feet by 40 feet