Price of maximum profit for packs of biscuits

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The company CarpiFoods sells packs of biscuits. At a price of $8$ € per pack it would sell $200$ packs a week; at a price of $5$ € it would sell $600$. The production cost of one pack is $3$ €.

• (a) find how the number of packs sold depends on the price (linear dependence);
• (b) write the formula for the weekly profit as a function of the price;
• (c) find the price that maximizes the profit and the value of the maximum profit.

Solution

(a) Demand $q(p)$ is a line through $(8;200)$ and $(5;600)$: $m=\frac{600-200}{5-8}=-\frac{400}{3}, \qquad q(p)=-\frac{400}{3}p+\frac{3800}{3}=-\frac{400}{3}\,(p-9.5).$ (b) Profit is (unit margin) $\times$ (quantity): $G(p)=(p-3)\,q(p)=-\frac{400}{3}\,(p-3)(p-9.5).$ It is a parabola opening downward, vanishing at $p=3$ and $p=9.5$. (c) The maximum is at the vertex, halfway between the zeros: $p^*=\frac{3+9.5}{2}=6.25\ \text{€}.$ At that price $q(6.25)=\dfrac{1300}{3}\approx433$ packs, and $G(6.25)=(6.25-3)\cdot\frac{1300}{3}=\frac{4225}{3}\approx1408.33\ \text{€/week}.$ $\boxed{\ p^*=6.25\ \text{€}, \qquad G_{\max}\approx1408.33\ \text{€/week}\ }$