a property owner wants to build a rectangular enclosure around some land that is next to the lot of a neighbor who is willing to pay for half the fence that actually divides the two lots. If the area is A, what should the dimensions of the enclosure be so that the cost to the owner is a minimum.
Dont really know Where to start with this one.
I DON'T GET IT!?
To find the dimensions of the enclosure that will minimize the cost to the owner, we need to break down the problem and find an equation representing the cost as a function of the enclosure's dimensions.
Let's assume that the land the property owner wants to enclose has length L and width W, and the neighbor's lot is adjacent to one side of the enclosure. The length of the shared fence with the neighbor's lot will be L (since L is also the length of the enclosure in that direction), while the other side will be W.
Now, let's consider the cost. The cost of the fence will depend on two factors: the length of the fence and the cost per unit length. Since the neighbor is willing to pay for half the fence that divides the two lots, the cost to the owner will be reduced by half.
Let's assume the cost per unit length of fence is C. The total cost to the owner will then be (L/2 + W) * C. We need to find the dimensions L and W such that this cost is minimized.
To minimize the cost, we can take the derivative of the cost function with respect to L and then with respect to W, and set them both equal to zero. This will give us the dimensions that minimize the cost.
Differentiating (L/2 + W) * C with respect to L gives us: (1/2) * C. Differentiating with respect to W gives us C.
Setting both derivatives equal to zero and solving for L and W gives us:
(1/2) * C = 0, and C = 0.
Obviously, this doesn't make sense since we can't have C = 0. This means that there is no minimum cost for the enclosure.
Therefore, in order to minimize the cost to the property owner, we cannot determine specific dimensions for the enclosure.
if the area enclosed is x by y, and the neighbor will pay for half the cost of the y side,
A = xy
C = 2x + 3/2 y
C = 2x + 3/2 (A/x)
dC/dx = 2 - 3A/(2x^2)
dC/dx = 0 when 3A/(2x^2) = 2
x = √(3A)/2
so, the lot is √(3A)/2 by 2√(A/3)
cost is thus √(3A) + 3/2 * 2√(A/3) = 2√(3A)