Arectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 648 feet of fencing is used. find the dimensions of the playground that maximize the total enclosed area.

A rectangular playground of sides a and b would have an area of a*b. The perimeter of the fence plus one side is 648ft; this can be written as 3a+2b. We can write b in the expression for the area as (648-3a)/2. The area is equal to (648a-3a^2)/2; the maximum value can be found by deriving the area expression and equate it to 0. The derivative is 324-3a; the value of the area is maximum when a=108ft. The dimensions that maximize the playground are a=108ft and b=162ft.

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0

answer: the function g(x) has the greatest slope.