- The profit function is given by \(p(x) = -10x^2 + 498x - 1,500\).
- Substitute the price \(x = 20\) into the function to get \(p(20) = -10(20)^2 + 498(20) - 1,500\).
- Calculate the values to find \(p(20) = -4,000 + 9,960 - 1,500 = 4,460\).
- The final profit is \(\boxed{\$4,460}\).
Explanation
- Analyzing the Profit Function
The company's monthly profit is modeled by the function \(p(x) = -10x^2 + 498x - 1,500\), where \(x\) represents the price per sign in dollars. We are tasked with finding the monthly profit when each sign is priced at \(x = \$20\). Our approach involves substituting \(x = 20\) into the given profit function and evaluating the expression to determine the resulting profit.
- Calculating the Monthly Profit
Substituting \(x = 20\) into the profit function \(p(x)\), we have:
\[p(20) = -10(20)^2 + 498(20) - 1,500\]
First, evaluate the squared term:
\[(20)^2 = 400\]
Next, multiply the terms:
\[-10(400) = -4,000\]
\[498(20) = 9,960\]
Now, combine these results:
\[p(20) = -4,000 + 9,960 - 1,500\]
Performing the addition and subtraction:
\[-4,000 + 9,960 = 5,960\]
\[5,960 - 1,500 = 4,460\]
The calculated monthly profit is \(\$4,460\).
- Conclusion and Option Selection
Comparing our result of \(\$4,460\) with the provided options:
- \(\$14,420\)
- \(\$4,460\)
- \(\$1,402\)
- \(\$18,020\)
The calculated profit corresponds to the second option. Thus, the company's profit when selling each sign for \(\$20\) is \(\$4,460\). The correct choice is \(\boxed{\text{\$4,460}}\).
Examples
Businesses often use profit models to determine the optimal pricing for their products. By evaluating a profit function at different price points, a business owner can estimate potential earnings and make informed decisions about pricing strategies to maximize revenue. For instance, if a local bakery sells custom cakes, they might model their profit based on the cost of ingredients and the price at which they sell each cake, helping them identify the most profitable price range.