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Janine babysits for $14.50 per hour. She also works as a dishwasher at her family's restaurant for $9.50 per hour. Her family needs her to be at the restaurant at least 7 hours per week, but she can't work there more than 10 hours per week. She also does not want to work more than a total of 15 hours each week. She wants to make at least $140 per week.
Write a system of inequalities that represents the constraints on Janine's part-time jobs. Explain what your inequalities represent in the context of the problem.
Janine wants to earn as much money as possible next week. What combination of hours should she spend at each job to do this? Justify your reasoning. ​

Respuesta :

Answer:

1) The system of inequalities are

14.5·x + 9.5·y  ≥ 140

7 ≤ y ≤ 10

x + y ≤ 15

2) 14.5·x + 9.5·y  ≥ 140 represents the total amount of money Janine can earn

7 ≤ y ≤ 10 represents the range of values, Janine can spend dishwashing

x + y ≤ 15 represents the total number of hours Janine will like to work each week

3) 8 hours babysitting, 7 hours dishwashing

Step-by-step explanation:

The given parameters are;

The amount per hour Janine makes from babysits = $14.50

The amount per hour Janine makes from dishwashing = $9.50

The minimum number of hours Janine can spend dishwashing = 7 hours

The maximum number of hours Janine can spend dishwashing = 10 hours

The maximum number of hours Janine can work each week = 7 hours

The minimum amount she wants to make each week = $140

Let x represent the number of hours Janine spends babysitting and let y represent the number of hours Janine spends dishwashing

1) From the question, we have;

14.5·x + 9.5·y  ≥ 140

7 ≤ y ≤ 10

x + y ≤ 15

2) Where

14.5·x + 9.5·y  ≥ 140 represents the total amount of money Janine can earn

7 ≤ y ≤ 10 represents the range of values, Janine can spend dishwashing

x + y ≤ 15 represents the total number of hours Janine will like to work each week

Making, y, the subject of the formula of the above inequalities and plotting as functions is given as follows;

y ≥ 140/9.5 - (14.5/9.5)·x

y ≤ 15 - x

3) In order to earn as much money as possible given that the amount Janine earns from babysitting is more than the amount she earns from dishwashing, Janine should spend the least amount of time dishwashing, which is 7 hours, as given, and then spend the remaining 8 hours babysitting to receive $14.5 × 8 + $9.5×7 = $182.5

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MrRoyal

Janine should work as a dishwasher for 7 hours, and as a babysitter for 8 hours, to earn as much money

Represent the hours babysitting with b, and the hour dishwashing with d.

So, we have the following systems of inequalities

  • [tex]14.5b + 9.5d \ge 140[/tex] ---her earnings
  • [tex]7\le d \le 10[/tex] -- the time she can spend at the restaurant
  • [tex]b + d \le 15[/tex] --- her weekly work hours

Next, we plot the graphs of the inequalities that represent the hours she can work.

i.e. the graphs of [tex]7\le d \le 10[/tex] and [tex]b + d \le 15[/tex]

From the attached graph, we have the following possible values

[tex](b,d) =(5,10) (8,7)[/tex]

Substitute these values in [tex]14.5b + 9.5d \ge 140[/tex]

So, we have:

At (5,10):

[tex]Earnings = 14.5 \times 5 + 9.5 \times 10[/tex]

[tex]Earnings = 167.5[/tex]

At (8,7):

[tex]Earnings = 14.5 \times 8 + 9.5 \times 7[/tex]

[tex]Earnings = 182.5[/tex]

The earnings at point (8,7) is greater than (5,10).

Hence, she should work as a dishwasher for 7 hours, and as a babysitter for 8 hours, to earn as much money

Read more about inequalities at:

https://brainly.com/question/18881247

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