Introduction: Production, Price, Demand, Revenue, \& Profit
A technology startup's market research department is tasked with determining the market viability of a new smartphone device. After suitable testing on the interest in a new smartphone, the research department determines the following price-demand equation:
\[
x=3.2 \times 10^{6}-500 p
\]
where $x$ is the amount of units (smartphones) in demand at price $p$ (in dollars).
For example, if the price of the new smartphone is set at $p=\$ 100$, then the amount of new smartphones in demand should be:
\[
x=3.2 \times 10^{6}-500(100)=3150000 \text { units }
\]
In addition, the financial department provides the cost function measured in dollars:
\[
C(x)=85 x+50000
\]
where $x$ is the nutnber of smartphones produced. Note that $\$ 50000$ is the fixed costs of production (maintenance, overhead, etc.) and $\$ 85$ is the cost (labor, materials, marketing, transportation, storage, etc.) per smartphone.
MTH 243 Calculus for Mgmt Life/Social Sci Midterm Formal Lab
Page 2 of 3
4. (10pts) The break-even points occur at some production level(s) $x$ such that the revenue $R(x)$ is equal to the cost $C(x)$. That is, the break-even points $x$ occur such that $R(x)=C(x)$. Use your work in the previous problems to find these break-even points. Hint: use the Quadratic Formula to solve your equation.
Previous work $R^{\prime}(x)=6400-0.004 x$

Step 1 ：Given the price-demand equation \(x=3.2 \times 10^{6}-500 p\), where \(x\) is the amount of units (smartphones) in demand at price \(p\) (in dollars).

Step 2 ：For example, if the price of the new smartphone is set at \(p=\$ 100\), then the amount of new smartphones in demand should be \(x=3.2 \times 10^{6}-500(100)=3150000 \text { units }\).

Step 3 ：The cost function provided by the financial department is \(C(x)=85 x+50000\), where \(x\) is the number of smartphones produced. Note that \(\$ 50000\) is the fixed costs of production (maintenance, overhead, etc.) and \(\$ 85\) is the cost (labor, materials, marketing, transportation, storage, etc.) per smartphone.

Step 4 ：The break-even points occur when the revenue equals the cost, i.e., when \(R(x) = C(x)\). The revenue function \(R(x)\) is the product of the price \(p\) and the demand \(x\), i.e., \(R(x) = px\).

Step 5 ：From the price-demand equation, we can express \(p\) in terms of \(x\), i.e., \(p = (3.2 \times 10^6 - x) / 500\). Substituting this into the revenue function gives \(R(x) = x(3.2 \times 10^6 - x) / 500\).

Step 6 ：Setting this equal to the cost function \(C(x) = 85x + 50000\) and solving for \(x\) will give us the break-even points. We can use the quadratic formula to solve this equation.

Step 7 ：The break-even points occur at production levels \(x \approx 7.92\) and \(x \approx 3157492.08\). In other words, the company breaks even when it produces and sells approximately 8 or approximately 3,157,492 units of the smartphone. These are the points at which the revenue from selling the smartphones equals the cost of producing them.

Step 8 ：\(\boxed{\text{Final Answer: The break-even points occur at production levels } x \approx 7.92 \text{ and } x \approx 3157492.08}\)