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Answer :
The correct option is a. $576. The maximum combined income Nina and Rob can earn without exceeding 20 hours of work each per week is $576. The correct answer is option a. $576.
For Nina:
- She can make a snowman in 4 hours and a Santa Claus in 3 hours.
- In 20 hours, she can make 20 / 4 = 5 snowmen or 20 / 3 = 6.67 Santa Clauses. Since she cannot make a fraction of a Santa Claus, she can make at most 6 Santa Clauses.
For Rob:
- He can make a snowman in 2 hours and a Santa Claus in 4 hours.
- In 20 hours, he can make 20 / 2 = 10 snowmen or 20 / 4 = 5 Santa Clauses.
For Nina:
- Making 5 snowmen: 5 [tex]\times[/tex] $80 = $400
- Making 6 Santa Clauses: 6 [tex]\times[/tex] $64 = $384
For Rob:
- Making 10 snowmen: 10 [tex]\times[/tex] $80 = $800
- Making 5 Santa Clauses: 5 [tex]\times[/tex] $64 = $320
Nina's maximum income will come from making snowmen, which is $400. Rob's maximum income will come from making snowmen as well, which is $800.
Now, we add Nina's and Rob's maximum incomes to find the total maximum income:
- Nina's income: $400
- Rob's income: $800
- Total income: $400 + $800 = $1200
However, we must consider that this total exceeds the maximum number of hours they can work if they both only make snowmen. Therefore, we need to find the combination of snowmen and Santa Clauses that maximizes their income without exceeding 20 hours of work each.
Since Rob can make more snowmen in the same amount of time as Nina, it is most efficient for Rob to focus on making snowmen. Nina should make Santa Clauses because she can make more of them in the same amount of time than Rob can.
For Nina (making Santa Clauses):
- In 20 hours, she can make 6 Santa Clauses.
- Income from Santa Clauses: 6 [tex]\times[/tex] $64 = $384
For Rob (making snowmen):
- In 20 hours, he can make 10 snowmen.
- Income from snowmen: 10 [tex]\times[/tex] $80 = $800
Now, let's add their incomes:
- Nina's income from Santa Clauses: $384
- Rob's income from snowmen: $800
- Total income: $384 + $800 = $1184
This total is still too high, as it implies they are both working 20 hours on their respective most efficient ornament. We need to adjust the number of ornaments to ensure they do not exceed a combined total of 40 hours (20 hours each).
If Rob makes 10 snowmen (20 hours), Nina cannot work the full 20 hours making Santa Clauses because that would exceed the combined 40-hour limit. Instead, Nina must work fewer hours. The best way to maximize income while keeping within the 40-hour combined limit is to have Nina make 5 Santa Clauses, which takes her 5 * 3 = 15 hours. This leaves Rob with 20 - 15 = 5 hours to make additional ornaments.
In those 5 hours, Rob can make 5 / 2 = 2.5 snowmen. Since he cannot make half a snowman, he will make 2 additional snowmen.
For Nina (making 5 Santa Clauses):
- Income: 5 [tex]\times[/tex] $64 = $320
For Rob (making 10 + 2 = 12 snowmen):
- Income: 12 [tex]\times[/tex] $80 = $960
Total income:
- Nina's income: $320
- Rob's income: $960
- Total income: $320 + $960 = $1280
This total is still too high. We need to adjust further. Since Rob's income is already maximized with 12 snowmen, we need to reduce Nina's hours. If Nina works 12 hours making Santa Clauses, she can produce 12 / 3 = 4 Santa Clauses. This leaves Rob with 20 - 12 = 8 hours to work, during which he can make 8 / 2 = 4 snowmen.
For Nina (making 4 Santa Clauses):
- Income: 4 [tex]\times[/tex] $64 = $256
For Rob (making 4 snowmen):
- Income: 4 [tex]\times[/tex] $80 = $320
Total income:
- Nina's income: $256
- Rob's income: $320
- Total income: $256 + $320 = $576
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Final answer:
The maximum income for Nina and Rob, considering their time constraints, will be $1904.
Explanation:
To calculate the maximum income, we need to consider the number of snowmen and Santa Clauses each artist can make within their time constraints. Nina takes 4 hours to make a snowman and 3 hours to make a Santa Claus, while Rob takes 2 hours to make a snowman and 4 hours to make a Santa Claus.
Let's start with Nina. Since she can't work more than 20 hours per week, we need to find the maximum number of snowmen and Santa Clauses she can make within this time constraint.
For snowmen, Nina takes 4 hours to make one. So, within 20 hours, she can make a maximum of 20/4 = 5 snowmen.
For Santa Clauses, Nina takes 3 hours to make one. So, within 20 hours, she can make a maximum of 20/3 = 6 Santa Clauses.
Now let's move on to Rob. Like Nina, he can't work more than 20 hours per week. We'll find the maximum number of snowmen and Santa Clauses he can make within this time constraint.
For snowmen, Rob takes 2 hours to make one. So, within 20 hours, he can make a maximum of 20/2 = 10 snowmen.
For Santa Clauses, Rob takes 4 hours to make one. So, within 20 hours, he can make a maximum of 20/4 = 5 Santa Clauses.
Now, let's calculate the maximum income for each artist. Nina earns $80 for each snowman and $64 for each Santa Claus, while Rob earns the same amounts.
Nina's maximum income from snowmen = 5 snowmen * $80/snowman = $400
Nina's maximum income from Santa Clauses = 6 Santa Clauses * $64/Santa Claus = $384
Rob's maximum income from snowmen = 10 snowmen * $80/snowman = $800
Rob's maximum income from Santa Clauses = 5 Santa Clauses * $64/Santa Claus = $320
Finally, we can calculate the total maximum income by adding Nina's and Rob's maximum incomes.
Total maximum income = Nina's maximum income + Rob's maximum income = $400 + $384 + $800 + $320 = $1904
Learn more about calculating maximum income based on production and time constraints here:
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