Colby and Danielle clean pools for extra money over the summer. Colby's income is determined by [tex]f(x)=3 x+12$[/tex], where [tex]$x$[/tex] is the number of hours. Danielle's income is [tex]$g ( x )=5 x +10$[/tex]. If Colby and Danielle were to combine their efforts, their income would be [tex]$h ( x )= f ( x )+ g ( x )$[/tex]. Assume Colby works 4 hours. Create the function [tex]$h ( x )$[/tex], and indicate if Colby will make more money working alone or by teaming with Danielle.
A. [tex]$h(x)=2 x+2$[/tex], team with Danielle
B. [tex]$h(x)=8 x+22$[/tex], work alone
C. [tex]$h(x)=2 x+2$[/tex], work alone
D. [tex]$h(x)=8 x+22$[/tex], team with Danielle
Answer (2)
Find the combined income function by adding Colby's and Danielle's income functions: h ( x ) = f ( x ) + g ( x ) = ( 3 x + 12 ) + ( 5 x + 10 ) = 8 x + 22 .
Calculate Colby's income working alone for 4 hours: f ( 4 ) = 3 ( 4 ) + 12 = 24 .
Calculate the combined income for 4 hours: h ( 4 ) = 8 ( 4 ) + 22 = 54 .
Since 24"> 54 > 24 , Colby will make more money teaming with Danielle. The combined income function is h ( x ) = 8 x + 22 , and Colby should team with Danielle.
Explanation
Understanding the Problem We are given Colby's income function f ( x ) = 3 x + 12 and Danielle's income function g ( x ) = 5 x + 10 . We need to find the combined income function h ( x ) = f ( x ) + g ( x ) and determine whether Colby makes more money working alone or teaming with Danielle if he works 4 hours.
Finding the Combined Income Function First, let's find the combined income function h ( x ) by adding f ( x ) and g ( x ) : h ( x ) = f ( x ) + g ( x ) = ( 3 x + 12 ) + ( 5 x + 10 ) Combining like terms, we get: h ( x ) = ( 3 x + 5 x ) + ( 12 + 10 ) = 8 x + 22
Calculating Colby's Income Alone Now, let's calculate Colby's income working alone for 4 hours. We substitute x = 4 into f ( x ) : f ( 4 ) = 3 ( 4 ) + 12 = 12 + 12 = 24 So, Colby makes $24 working alone for 4 hours.
Calculating the Combined Income Next, let's calculate the combined income for 4 hours. We substitute x = 4 into h ( x ) : h ( 4 ) = 8 ( 4 ) + 22 = 32 + 22 = 54 So, the combined income for 4 hours is $54.
Comparing Incomes To determine if Colby makes more money working alone or teaming with Danielle, we compare Colby's income alone, f ( 4 ) , with his share of the combined income. Since we are not given information about how the combined income is split, we will compare Colby's income alone with the total combined income. Since 24"> 54 > 24 , the combined income is greater than Colby's income alone. Therefore, Colby and Danielle make more money working together. The question asks if Colby will make more money working alone or teaming with Danielle. Since the combined income is $54, and Colby's income alone is $24, teaming with Danielle results in a higher total income.
Conclusion The combined income function is h ( x ) = 8 x + 22 . Since the combined income for 4 hours ($54) is greater than Colby's income alone for 4 hours ($24), Colby will make more money teaming with Danielle.
Examples
Understanding income models can help in various real-life scenarios. For example, if you and a friend are starting a car-washing business, you can model your earnings based on the number of cars you wash. By combining your efforts, you can determine if working together yields a higher income compared to working individually. This kind of analysis helps in making informed decisions about partnerships and resource allocation, ensuring that everyone benefits from the collaboration. The combined income function h ( x ) = 8 x + 22 helps to predict total earnings based on hours worked, facilitating better financial planning.
The combined income function is h ( x ) = 8 x + 22 . Colby earns $24 alone for 4 hours, while the combined income with Danielle for the same time is $54, meaning he makes more money working together. The correct choice is D: h ( x ) = 8 x + 22 , team with Danielle.
;
