Question 1142726
.
<pre>
The <U>PROFIT</U> function P(x) is the difference  REVENUE - COST 


    P(x) = 200x - (500000+80x+0.003x^2) = -0.003x^2 + 120x - 500000.     (1)



This quadratic function plot is opened downward, so it has the maximum.


The maximum of a quadratic function is achieved at  x = {{{-b/(2a)}}}  (referring to the general form f(x) = ax^2 + bx + c).


In our case  " a " = -0.003, b = 120,  so the maximum is achieved at  x = {{{-120/(2*(-0.003))}}} = {{{120/0.006}}} = 20000.


It is  LESS  than the company capacity of  30000.


So, the company profit will be maximum at x = 20000.


The profit value is then  

    {{{-0.003*20000^2 + 120*20000 - 500000}}} = 700,000.



<U>ANSWER</U>.  20,000 units should be manufactured and sold.
</pre>

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


On finding the maximum/minimum of a quadratic function &nbsp;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>



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 &nbsp;"<U>Finding minimum/maximum of quadratic functions</U>". 



================


Sorry, in my solution I used the standard &nbsp;Algebra &nbsp;approach instead of Calculus.


But you will get the same answer, &nbsp;using &nbsp;Calculus.


For it, &nbsp;simply differentiate the quadratic function &nbsp;P(x) &nbsp;(formula (1))  &nbsp;and equate the derivative to zero 



<pre>
    P'(x) = -0.003*2*x + 120 = 0


    120 = 0.006x


    x = {{{120/0.006}}} = 20000.       <U>ANSWER</U>
</pre>

Solved.    &nbsp;&nbsp;// &nbsp;&nbsp;So, &nbsp;now you have both &nbsp;Algebra &nbsp;and &nbsp;Calculus &nbsp;solutions.