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SunBright manufactures solar lanterns. The revenue and expense functions are given by:

$R(q)=500 q-10 q^2$

$E(q)=2,000+200 q$ where $q$ represents quantity.



Based on the revenue and expense functions provided, what is the correct profit function $P(q)$ for SunBright's solar lanterns?


A$P(q)=-10 q^2+500 q-2,000$
B$P(q)=-10 q^2+700 q+2,000$
C$P(q)=-10 q^2+300 q-2,000$
D$P(q)=-10 q^2+700 q-2,000$

Verified Answer :

  • Define the profit function as the difference between the revenue and expense functions: \(P(q) = R(q) - E(q)\).
  • Substitute the given functions: \(P(q) = (500q - 10q^2) - (2,000 + 200q)\).
  • Distribute the negative sign: \(P(q) = 500q - 10q^2 - 2,000 - 200q\).
  • Combine like terms to arrive at the final profit function: \(P(q) = -10q^2 + 300q - 2,000\), which corresponds to option \(\boxed{c}\).

Explanation

  1. Understanding the Problem and Data Analysis

The problem asks to determine the profit function \(P(q)\) for SunBright's solar lanterns. Profit is defined as the difference between the total revenue, \(R(q)\), and the total expenses, \(E(q)\). Given are the functions:
\[R(q) = 500q - 10q^2\]
\[E(q) = 2,000 + 200q\]
To find the profit function, we use the formula:
\[P(q) = R(q) - E(q)\]

  1. Setting Up the Profit Function

Now, we substitute the expressions for \(R(q)\) and \(E(q)\) into the profit formula:
\[P(q) = (500q - 10q^2) - (2,000 + 200q)\]
To simplify this expression, we distribute the negative sign across the terms within the parentheses for the expense function \(E(q)\):
\[P(q) = 500q - 10q^2 - 2,000 - 200q\]

  1. Simplifying the Expression

Next, we group the terms with \(q\):
\[P(q) = -10q^2 + (500q - 200q) - 2,000\]
By subtracting \(200q\) from \(500q\), we get \(300q\). Substituting this back into the expression, we obtain:
\[P(q) = -10q^2 + 300q - 2,000\]

  1. Conclusion and Selection of the Correct Option

Comparing our resulting profit function \(P(q) = -10q^2 + 300q - 2,000\) with the given options:

A\(P(q) = -10q^2 + 500q - 2,000\)
B\(P(q) = -10q^2 + 700q + 2,000\)
C\(P(q) = -10q^2 + 300q - 2,000\)
D\(P(q) = -10q^2 + 700q - 2,000\)

The derived function matches option (c). Thus, the correct profit function is \(P(q) = -10q^2 + 300q - 2,000\). Final answer is \(\boxed{c}\).

Examples

In business, calculating profit is essential for understanding the viability of a product. If a company knows how their revenue changes with production volume (economy of scale or market saturation) and how their costs behave (fixed costs vs. variable costs), they can model their profit as a function. This allows managers to find the "break-even point" (where \(P(q)=0\)) or the "maximum profit point" by analyzing the peak of the quadratic function, helping them decide the optimal quantity to manufacture.

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