Let f(x) be the perimeter of a rectangle with an area 16units2 and one side with length x. f(x) = What is the minimum perimeter of all rectangles with this area? Perimeter =

Answer 1

This is a minimization problem - we will figure out the x which minimizes the perimeter.

Area = 16 = x * (unknown side)

So, the unknown side, in terms of x, is (16/x).

Now we have to write a function for the perimeter, which is the sum of all the sides. But now we know all the sides - two of them are of length x, and the other two are (16/x).

For the perimeter, we then have:

P(x) = (32 / x) + (2x)

Now we have to find the minimum of this function - the minimum will be when the derivative of the function is zero. This function has one extreme, which is the minimum we will find.

dP(x) / dx = 0 = - 32 / x^2 + 2

And now we rearrange:

32/x^2 = 2

So x^2 = 16, and x=4. Note, for a rectangle with area 16 and one side length of 4, this actually has to be a square. And the minimum perimeter is 4 sides x 4 (units) = 16 units.

Enjoy ^u^