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CONCORDIA UNIVERSITY MATH 209 – APRIL 2024 FINAL EXAM

QUESTION NO. 7
(8 marks) A company manufactures and sells 𝑥 items per week. The weekly cost function and price-demand equations are, respectively,

\begin{matrix} C (x) = 60 x + 42000 \\ p = 200 - \frac{x}{40} \end{matrix}

where 𝑥 is the weekly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government decides to tax the company $10 for each item it produces.
Find the production level and price per item that produce the maximum profit.
What is the maximum profit?

Solution:
In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).
We will use the equation that relates Profit with Revenue and Cost:

\begin{matrix} Profit = Revenue - Cost \\ P (x) = R (x) - C (x) \end{matrix}

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $2 tax on each item produced. This means we need to add 2𝑥 to the cost function:

\begin{matrix} C (x) = 60 x + 42000 + 10 x \end{matrix}

The new cost function then becomes:

\begin{matrix} C (x) = 70 x + 42000 \end{matrix}

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

\begin{matrix} R (x) = p \cdot x \end{matrix}

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

\begin{matrix} R (x) = (200 - \frac{x}{40}) \cdot x \\ R (x) = 200 x - \frac{x^{2}}{40} \end{matrix}

Now that we have the Revenue function, we can substitute it into the Profit function

and get a Profit function in terms of 𝑥.

\begin{matrix} P (x) = R (x) - C (x) \\ P (x) = (200 x - \frac{x^{2}}{40}) - (70 x + 42000) \\ P (x) = 200 x - \frac{x^{2}}{40} - 70 x - 42000 \\ P (x) = - \frac{1}{40} x^{2} + 130 x - 42000 \end{matrix}

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 2600.

Price per item:

The price of each item can be calculated using the price function given in the question:

\begin{matrix} p = 200 - \frac{x}{40} \\ Substitute 𝑥=928 in for 𝑥. \\ p = 200 - \frac{2600}{40} \\ p = 200 - 65 \\ p = 135 \end{matrix}

So the price that maximizes the profit is $135.

To find the maximum monthly profit, we substitute 𝑥 = 135 into the Profit and calculate this maximum value:

\begin{matrix} P (x) = - \frac{1}{40} x^{2} + 130 x - 42000 \\ P (928) = - \frac{1}{40} (2600)^{2} + 130 (2600) - 42000 \\ P (2600) = 127000 \end{matrix}

So the maximum monthly profit is $127000.

Final Answer:

The production level that maximizes profit is 2600 items per week, with a corresponding price of $135 per item. The maximum profit is $127000 per week.