let 'x' and 'y' represent the dimensions of the rectangle

s = the distance from one corner to the midpoint of a non-adjacent side

A rectangle is to have an area of 64 cm squared means:

xy = 64 divide both sides by x

y = 64/x

Using the Pythagorean Theorem we can write for s:

s^2 = x^2 + (y/2)^2

plug y = 64/x into the last equation:

s^2 = x^2 + ((64/x)/2)^2

s^2 = x^2 + (32/x)^2

If we plot the function we see that it has minimum.

Click here to see the graph of the function: y = (x^2 + (32/x)^2)^(1/2)

Click to see all the steps

We wish to minimize s and hence we need to differentiate s^2 = x^2 + (32/x)^2 with respect to x:

2s(ds/dx) = 2x - (2*32)/(x^3)

When s is a relative minimum, ds/dx = 0 that is

2x - (2*32)/(x^3) = 0

by solving we find:

x^4 = 32

x = 2.38 cm

click here to see the step by step solution of the equation:

Click to see all the steps

y = 64/2.38

y = 29.89 cm

the dimensions of the rectangle are 2.38 cm and 29.89 cm.