Question

. A firm produces a commodity and receives $100 for each unit sold. The cost of producing and selling x units is 20x + 0.25x 2 dollars. (a) Find the production level which maximizes profit. (in other words, find the x that maximizes profit) (b) A tax of $10 per unit is imposed on the firm. What is the new optimum production level?

Answer 1

So for each unit they get $100, so if they sell "x" units, they intake 100x, that's just the gross income, and therefore the Revenue.

Now, they have a cost of 20x + 0.25x² if they "x" units, now, bearing in mind that if you pluck out the costs out of the incoming revenue, what's leftover, is a surplus amount, namely the Profit.  Therefore, profit p(x) = revenue - costs, or  100x - (20x + 0.25x²), which is p(x) = -0.25x² +80x.

a)
now, what's the highest profit they can make?  check the picture below.

$\bf p(x)=-0.25x^2+80x\\\\ -------------------------------\\\\ \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lcclll} p(x) = &{{ -0.25}}x^2&{{ +80}}x&{{ +0}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right) \\\\\\ \left( -\cfrac{80}{-0.50}~,~0-\cfrac{80^2}{-1} \right)\implies \left( \stackrel{\textit{quantity}}{160}~,~\stackrel{\textit{profit value}}{6400} \right)$

b)

well, if a tax is imposed on them, and say, they take it out of the revenue, then their new revenue is not 100x, is 90x, thus   $\bf p(x)=-0.25x^2+70x$

and you can use the same formula as above to get the x-coordinate for that vertex.

$\bf p(x)=-0.25x^2+70x\\\\ -------------------------------\\\\ \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lcclll} p(x) = &{{ -0.25}}x^2&{{ +70}}x&{{ +0}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right) \\\\\\ \left( -\cfrac{70}{-0.50}~,~0-\cfrac{70^2}{-1} \right)$