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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.

Respuesta :

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.

The conclusions are as follow;

a.  The area of the rectangle is A = x(400 - 4x).

b.  The domain of the function is (0, 100).

c.  The dimension of the swimming section is 30 ft to maximize the area.

What is Geometry?

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

A summer camp has 400 ft of float line (L).

Let inner sections length be x.

a.  The amount of the side length of the rectangular can be calculated from the total length (y) given as;

[tex]\rm y = L -x-x-x-x\\\\y = L - 4x\\\\y = 400- 4x[/tex]

The area of the rectangle will be

A = x × y

A = x(400 - 4x)

b.  The domain of the function A(x) will be

400x - 4x² > 0

4x(100 - x) > 0

Then x > 0

and x < 100

Then the domain of the function is (0, 100)

c.  From the graph given, we can conclude that the area is maximum when x = 30 ft. denoted by the turning point of the graph.

Thus, the dimension of the swimming section is 30 ft to maximize the area.

More about the geometry link is given below.

https://brainly.com/question/7558603

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