Question

A. A rectangular garden with perimeter 40m is to be fenced.
1- What are dimensions of a garden and its maximum area?
2-what are the dimensions of a rectangle with area 45m square. and the minimum perimeter?

Answer 1

Perimeter of rectangle is 2L + 2w : where L,w are length and width
Area is L*w
ok so 2L + 2w = 40 and we need to maximize area
This means define Area as a function of one variable, take derivative and set it equal to zero.
Lets solve for L
-> L = (40-2w)/2 = 20 - w
Now substitute (20-w) in for L in Area formula
->A = (20-w)*w = 20w - w^2
Differentiate using power rule
->dA/dw = 20 - 2w
set =0 and solve
-> 20-2w = 0 -> 2w=20 -> w =10
Now plug that value in for w in our perimeter formula and solve for L
->2L + 2(10) = 40 -> 2L = 20 -> L = 10
So L=w=10 and area = 10*10=100

Hope this helps