Question

A summer camp has 400 ft of float line with which to rope off three adjacent rectangular areas of a lake for swimming? lessons, one for each of three levels of swimming ability. A beach forms one side of the swimming areas. Suppose that the width of each area is x feet. Answer parts? a) through ?c). Three rectangles each with a vertical side of length 'x' but with varying horizontal lengths are situated adjacent to each other.

Answer 1

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.