Question 1061030
.
A cable television firm presently serves 6,300 households and charges $14 per month. A marketing survey indicates that 
each decrease of $1 in the monthly charge will result in 630 new customers. Let R(x) denote the total monthly revenue 
when the monthly charge is x dollars. Find the value of x that results in maximum monthly revenue. 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
Let C(x) = 14 -x be the monthly charge for one single customer as the function of the projected decrease of "x" dollars.

Let N(x) = 6300 + 630x be the number of customers as the function of the same variable: the projected decrease of "x" dollars.

Then monthly revenue R is the product R = C*N, or

R = (14-x)*(6300 + 630x),    (1)

and we need to find the maximum of this function.

Write the function (1) in the general form for the quadratic function

R(x) = {{{14*6300 - 6300x + 14*630x - 630x^2}}},   or

R(x) = {{{-630x^2 + 2520x + 88200}}}.


       Now, let me remind you that for the general quadratic function f(x) = {{{ax^2 + bx + c}}} with the negative coefficient a < 0  
       the theory predicts the maximum  at x = {{{-b/(2a)}}}.


In our case the maximum will be at  x = {{{-2520/(2*(-630))}}} = 2.

It means that the maximum is predicted at the $2 dollars decreased charge of $14 - $2 = $12.

The number of customers then will be 6300 + 2*630 = 7560,  and the total revenue will be $12*7560 = $90720.

       You can check that this revenue is the same as calculated in accordance with the function R(x) = {{{-630x^2 + 2520x + 88200}}}  at   x = 2:
       R(2) = {{{-630*2^2 + 2520*2 + 88200}}} = 90720.

<U>Answer</U>.  The value of monthly charge that results in maximum monthly revenue is $12 dollars. 
</pre>

See the plot below where the revenue R(x) is shown as the function of projected <U>decrease of charge</U>:


{{{graph( 330, 330, -1.5, 5.5, -10000, 100000,
          -630*x^2 + 2520*x + 88200
)}}}


Plot y = R(x) = {{{-630*x^2 + 2520*x + 88200}}}


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>


Also, 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>".