Fisrt let's formulate an equation based on what the word problem is giving us.

The lenght and width of a rectangle have a sum of 78= L + W=78

Then it's asking what dimensions give the maximum area, so you must know your Area equation=

A= L(W)

Solve:

L+W=78

L=78-W

Now plug this into your area equation

A= (78-W) (W)

A=78W-W^2

Now a very interesting concept can be applied, we need to take the derivative of this equation, and must set A= 0.

0= 78(1)-2W

*The (1) came to be because the derivative of a variable is 1. W^2 was changed to 2w because of the power rule of derrivatives, thsis sttes that we can bring out the power in front of the variable and subtract 1 from the exponent so we get 2w^1 which is 2W*

Now move things around and solve for W:

-78=-2W

W=39

Now plug in value back into the L=78-W equation to solve for L

L= 78- 39

L=39

Hope this Helps :)