Answer:
a) A = x*(400 - 4*x)
b) domain of function A(x) is ( 0 , 100 )
c) dimension of swimming section x = 30 ft maximizes area.
Step-by-step explanation:
Given:
- Total length of float-line used L = 400 ft
- Inner sections length x
Find:
a) Express the total area A as a function of x
b) Find the domain of the function
c) Using the graph, find the dimensions that leads to largest area
Solution:
- The amount of side length of the rectangle can be calculated from the total length given y:
y = L - x - x - x - x
y = L - 4*x
y = 400 - 4*x
- The area of a rectangle is as follows:
A = x*y
- Replace y with the expression derived first:
A = x*(400 - 4*x)
- To find the domain of the function we know that A >= 0:
400*x - 4x^2 > 0
x(400 - 4x) > 0
x > 0 , 400 - 4*x < 0
x < 100
- Hence, the domain of function A(x) is ( 0 , 100 )
- From the graph given, we can see that Area is maximum when x = 30 ft. Denoted by the turning point of the graph.
- Hence, the dimension of swimming section x = 30 ft maximizes area.
