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(z)=-10 z^2+400 z-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 the value of z = 20 into the profit function p ( z ) = − 10 z 2 + 400 z − 1500 .
Calculate p ( 20 ) = − 10 ( 20 ) 2 + 400 ( 20 ) − 1500 .
Simplify the expression: p ( 20 ) = − 4000 + 8000 − 1500 = 2500 .
The profit is 2500 .
Explanation
Understanding the Problem The problem provides a profit function p ( z ) = − 10 z 2 + 400 z − 1500 , where z is the price of each sign. We need to find the profit when the price of each sign is $20 .
Substituting the Value To find the profit when the price is $20 , we substitute z = 20 into the profit function: p ( 20 ) = − 10 ( 20 ) 2 + 400 ( 20 ) − 1500
Calculating the Profit Now, we calculate the value: p ( 20 ) = − 10 ( 400 ) + 8000 − 1500 p ( 20 ) = − 4000 + 8000 − 1500 p ( 20 ) = 4000 − 1500 p ( 20 ) = 2500
Finding the Correct Answer The company's profit if it sells each sign for $20 is $2500 . However, this is not one of the options provided. Let's re-examine the options and the calculations to ensure accuracy. It seems there was a mistake in the provided options. The correct profit is $2500 .
Examples
Understanding profit functions is crucial for businesses. For instance, a local bakery can use a similar function to model their profit based on the price of each cake they sell. By analyzing this function, they can determine the optimal price to maximize their profit. This helps in making informed decisions about pricing strategies and overall business planning, ensuring they remain competitive and profitable in the market.
The company's profit when selling each sign for $20 is $2500. This result was obtained by substituting z = 20 into the profit function and simplifying the expression. However, this profit value is not in the provided answer choices.
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