SOLUTION: Good Evening, the question I am having a hard time with is this: A car dealership has 24ft of dividers with which to enclose a rectangular play space in a corner of a customer lo

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Question 1165391: Good Evening, the question I am having a hard time with is this:
A car dealership has 24ft of dividers with which to enclose a rectangular play space in a corner of a customer lounge. The sides against the wall require no partition. Suppose the play space is x feet long.
a) Express the area of the play space as a function of x
b) Find the domain of the function
c) Using the graph shown below, determine the dimensions that yield the maximum area ( the graph here is curved from (0,0) to (24,0) with its highest point at (12, 144).
I have figured out part A already and created the equation A(x)=24x-x^2 but I am having a hard time with parts b and c. If you could help me understand it a little better I would greatly appreciate it. Thank you! :)

Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!

The domain of any polynomial function is the set of real numbers, but in this case, any value of the independent variable that makes the value of the function negative is absurd and should therefore be excluded from the domain. The graph of your function is a parabola that opens downward and the portion of the graph that is above the -axis is the portion of the function that is positive. Find the two zeros of your function and the domain is the OPEN interval between the two zeros. The vertex of the parabola is at the value of that provides the maximum area and the value of the function at that point is the value of that maximum area.

In general, for a rectangle with a given perimeter, a square yields the maximum area.

John

My calculator said it, I believe it, that settles it

I > Ø

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

            The other tutor misread the problem,

            So I came to bring a correct solution.

Your start was incorrect. If x is the length of the play space (and we consider it as the length of the rectangle), then the width of the rectangle is . Therefore, the formula for the area of this rectangular play space is A(x) = = = 12x - . It is CONSISTENT with your plot/figure, which is the parabola with the zeros (x-intersections) at x= 0 and x= 24; and the maximum at the midpoint x= 12 between the x-intersections.

Solved.

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To see many other similar solved problems, look into the lesson
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area (*)
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
in this site.

Of these lessons, the most relevant to you is the lesson marked (*) in the list.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.