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Project $1$: Break$-$Even Analysis

In this project, we combine demand curves and cost curves with least$-$squares lines to determine break$-$even points. Remember that a "curve" can in fact be linear. Use technology $($Excel$,$ geogebra, graphing calculator$)$ to solve the following questions. Please be clear when answering each prompt. Include units and all relevant information to support your answers. All questions are weighted equally at $20$ points each.

$\backslash$table$[[q($thousands$),p($dollars$)],[100,360],[200,316],[300,288],[400,236],[500,200]]$

The data in the table can be used to obtain the demand curve for a monopolist who manufactures and sells a unique type of small, submersible, and nearly indestructible video camera. The first column gives several production quantities in thousands of cameras $(q),$ and the second column gives the corresponding prices per camera $(p)$ at that production level. For instance, in order to sell $200,000$ cameras, the manufacturer must set the price at $$316$ per camera. Find the least$-$squares line that best fits these data; that is$,$ find a demand curve for the camera.

What is the correlation coefficient and does it imply that the data seem to fit a linear model?

Use the demand curve to estimate the price that must be charged in order to sell $350,000$ cameras. Calculate the revenue for this price and quantity. Note: The revenue is the amount of money received from the sale of the cameras.

Use the demand curve to estimate the quantity that can be sold if the price is $$300$ per camera. Calculate the revenue for this price and quantity.

Determine the expression that gives the revenue from producing and selling $q$ thousand cameras.

Assuming that the manufacturer has fixed costs of $$8$ million and that the variable cost of producing each thousand cameras is $$100,000,$ find the equation of the cost curve.

Graph the revenue curve and the cost curve. Determine the point$($s$)$ of intersection.

What is the break$-$even point? That is$,$ what is the lowest value of $q$ for which cost equals revenue $($zero profit$)?$

For what values of $q$ will the company make a profit?

One misconception in sales is that the greatest profit occurs by producing and selling as many units as possible. How many thousands of cameras should be produced to

maximize profit in this example? That is$,$ when does it appear that the profit $($revenuecost$)$ is the greatest? Approximately, what is the total profit at that point?