Question 1195243
.
Please explain all the steps to my questions
1. The Sum of a number and three times another number is 18. 
Find the numbers if their product is maximum.
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<pre>
Let one number be x and another number be y.


    +---------------------------------------+
    |    We are given  x + 3y = 18,         |
    |                                       |
    |    so we can express  x = 18 - 3y.    |
    +---------------------------------------+


We are given that the product xy is maximum.


The product xy is

    xy = (18-3y)*y = 18y - 3y^2.


It is a quadratic function of variable y.

Since the coefficient at y^2 is negative  (it is -3), the plot of this quadratic function 
is a downward parabola, which clearly has the maximum.


Its maximum is nothing else as a vertex of the parabola.


To find the position of the maximum point (the position of vertex), there is a general formula for the argument of the quadratic function.


This general formula is  {{{y[max]}}} = {{{- b/(2a)}}},  where "a" is a coefficient at y^2

and b is the coefficient at y of the quadratic function.


In your case, a = -3;  b = 18.  Therefore, the position of the maximum {{{y[max]}}} on the coordinate line is

    {{{y[max]}}} = {{{-18/(2*(-3))}}} = {{{-18/(-6)}}} = 3.


Since we just know y = {{{y[max]}}} = 3, we can find x = 18 - 3y = 18 - 3*3 = 9.


Thus the answer to the problem is  x= 9;  y= 3.


The problem is just solved.
</pre>

The given explanation is a standard mantra to pronounce when solving such problems.


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On finding the maximum/minimum of a quadratic function see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>



Consider these lessons as your textbook, &nbsp;handbook, &nbsp;tutorials and &nbsp;(free of charge) &nbsp;home teacher.

Learn the subject from there once and for all.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


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



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.