Question 1194341
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Part (a)


Profit = Revenue - Cost
P(x) = R(x) - C(x)
P(x) = (75x-0.5x^2) - (35x+300)
P(x) = 75x-0.5x^2 - 35x-300
P(x) = -0.5x^2 + 40x - 300


The profit expression is -0.5x^2 + 40x - 300


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Part (b)


Set the profit expression equal to 300 and solve for x.


-0.5x^2 + 40x - 300 = 300
-0.5x^2 + 40x - 300 - 300 = 0
-0.5x^2 + 40x - 600 = 0
Factoring may be possible, but I prefer to use the quadratic formula.


Plug in a = -0.5, b = 40, c = -600
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(40)+-sqrt((40)^2-4(-0.5)(-600)))/(2(-0.5))}}}


{{{x = (-40 +- sqrt(400))/(-1)}}}


{{{x = (-40+- 20)/(-1)}}}


{{{x = (-40+20)/(-1)}}} or {{{x = (-40-20)/(-1)}}}


{{{x = (-20)/(-1)}}} or  {{{x = (-60)/(-1)}}}


{{{x = 20}}} or  {{{x = 60}}}


Another approach you can take is to graph these two equations
y = -0.5x^2 + 40x - 300 
y = 300
using a tool like Desmos
<a href = "https://www.desmos.com/calculator/jfde1smu5r">https://www.desmos.com/calculator/jfde1smu5r</a>
In the link above, the graph is shown. The points of intersection will give us the x solutions. Ignore the y coordinates of each point.
You can click on the intersection point to have the coordinates show up.
The intersections of (20,300) and (60,300) lead to the solutions found earlier (x = 20 and x = 60).


Interpretation: If the company sells either 20 items or 60 items, then they will have a profit of $300.
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