Question 1206524
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sammy is selling widgets. If he sells 1 widget, it costs him $1 to produce it and he can sell it for $10. 
If he sells 2 widgets, it costs him $2 to produce 2 widgets, but he can only get $9 for each widget. 
It costs $1 to produce each widget. The average price decreases by $1 for every extra widget sold. 
ow many widgets should sammy sell to maximise his profit?
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<pre>
Cost producing x widgets is x dollars.

Selling price of each one widget in the group of x widgets is 11-x.


So the profit is

    P(x) = Revenue(x) - Cost(x) = x*(11-x) - x = -x^2 + 10x.


The maximum of P(x) is at  {{{x[max]}}} = " {{{-b/(2a)}}} ",  where  "a"  is the coefficient
at  {{{x^2}}},  "b"  is the coefficient at x.


In this problem,  {{{x[max]}}} = {{{-10/(2*(-1))}}} = {{{10/2}}} = 5.


So, the optimum number of widgets in group to produce and to sell is 5.


It gives the maximum profit of -5^2 + 10*5 = -25 + 50 = 25 dollars per group.
</pre>

Solved.


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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>

in this site.