Select the correct answer.
A home-based sign company uses this function to model its monthly profit, where $x$ is the price of each sign it sells.
[tex]p(x)=-10 x^2+498 x-1,500[/tex]
What is the company's profit if it sells each sign for $20?
A. $1,402
B. $14,420
C. $4,460
D. $18,020
Answer (2)
Substitute x = 20 into the profit function p ( x ) = − 10 x 2 + 498 x − 1 , 500 .
Calculate p ( 20 ) = − 10 ( 20 ) 2 + 498 ( 20 ) − 1 , 500 .
Simplify the expression: p ( 20 ) = − 4000 + 9960 − 1500 = 4460 .
The company's profit is $4 , 460 .
Explanation
Understanding the Problem We are given the profit function p ( x ) = − 10 x 2 + 498 x − 1500 , where x is the price of each sign. We want to find the profit when the price of each sign is x = 20 .
Substituting the Value To find the profit when x = 20 , we substitute x = 20 into the profit function: p ( 20 ) = − 10 ( 20 ) 2 + 498 ( 20 ) − 1500
Calculating the Profit Now, we calculate the value: p ( 20 ) = − 10 ( 400 ) + 498 ( 20 ) − 1500
p ( 20 ) = − 4000 + 9960 − 1500
p ( 20 ) = 5960 − 1500
p ( 20 ) = 4460
Final Answer The company's profit when it sells each sign for $20 is $4 , 460 . Therefore, the correct answer is C.
Examples
Understanding profit functions can help businesses optimize their pricing strategies. For example, a local bakery can use a similar quadratic function to model their profit based on the price of a cake. By finding the price that maximizes the profit, they can make informed decisions about how to price their products to achieve the best financial outcome. This helps in planning production and managing costs effectively.
The company's profit when selling each sign for $20 is $4,460, calculated by substituting $20 into the profit function. Thus, the correct answer is C. $4,460.
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