SOLUTION: A window is to be constructed in the shape of an equilateral triangle on top of a rectangle if its perimeter is to be 600 cm, What is the maximum possible area of the window?

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Question 1146864: A window is to be constructed in the shape of an equilateral triangle on top of a rectangle if its perimeter is to be 600 cm, What is the maximum possible area of the window?
Answer by ikleyn(52784)   (Show Source): You can put this solution on YOUR website!
.

Let W be the width of the window, in centimeters.


Then the vertical length of the rectangular part of the window is L = 0.5*(600 - 3W) centimeters.


Thus the area of this special form window is


     A(w) = W*0.5*(600-3W) +   cm^2.     (1)


The first addend in the formula is the area of the rectangular part, while the second addend is the area of the triangular part.


Thus the area is this quadratic function of the variable "w"


    A(w) =  +  +  =  + 300*W.    (2)


They ask to find the maximum of this quadratic form.


For the general quadratic form f(x) = ax^2 + bx + c  with negative leading coefficient "a",  

the maximum is achieved at x = .


In our case,  a = ,  b = 300.



Therefore, the quadratic form achieves the maximum at


    W =  = 140.583 cm.


To find the maximum area, simply substitute this value of W into the formula (2)


     =  + 300*140.583 = 21087.41 cm^2.      ANSWER

Solved.

--------------------

On finding maximum/minimum of a quadratic functions and solving other similar minimax problems see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - 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
    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

A convenient place to observe all these lessons from the  "bird flight height"  is the last lesson in the list,  marked by  (*).

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.


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