Question

16m of fencing is used to construct a rectangular garden in the city park. The length of the rectangle is randomly chosen integral number in metres. if the garden is going to cost $65/m2, find the expected cost of the installation

Answer 1

We are given the total perimeter of the rectangular plot here as 16m. Therefore, given the length and breadth of the rectangle as X and Y respectively. Then, we have here:
X + Y = 16/2 = 8

X = 8 - Y

Area = XY

Expected cost is computed here as:
= 65 * Area

= 65XY

= 65X(8 - X)

Now as we know here that X ranges uniformly from 0 to 16 as the length could be anything from 0 to 16, therefore we have here:

Therefore the expected values here are computed as:

Now the second moment here is computed as:

Therefore the expected cost of the installation here is computed as:

E(65X(8 - X)) = 65*8*E(X) - 65E(X2) = 65*8*8 - 65*85.3333 = -1386.6667

Therefore -1386.6667 is the required expected cost of the installation here.