Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Certainly! Let's break down the steps to solve the problem step-by-step:
### Given Information:
- Lia must work at least 5 hours per week in the restaurant ([tex]$r \geq 5$[/tex]).
- She can work a maximum of 15 hours per week overall ([tex]$r + y \leq 15$[/tex]).
- She wants to earn at least [tex]$120 per week ($[/tex]8r + [tex]$12y \geq 120$[/tex]).
- Hourly wage at the restaurant is [tex]$8 per hour. - Hourly wage for yard work is $[/tex]12 per hour.
We'll answer each part of the question:
### 1. Graph the Inequalities:
To graph the inequalities:
- [tex]$r \geq 5$[/tex]:
Draw a vertical line at [tex]$r = 5$[/tex] and shade to the right (including the line).
- [tex]$r + y \leq 15$[/tex]:
Draw a line passing through points [tex]$(15, 0)$[/tex] and [tex]$(0, 15)$[/tex]. Shade below this line (including the line).
- [tex]$8r + 12y \geq 120$[/tex]:
Rearrange to [tex]$y \geq \frac{120 - 8r}{12} \Rightarrow y \geq 10 - \frac{2r}{3}$[/tex].
Draw this line by finding intercept points, let’s find the intercepts:
- Set [tex]$r = 0 \Rightarrow y = 10$[/tex]
- Set [tex]$y = 0 \Rightarrow $[/tex] [tex]$r = 15$[/tex]
Plot these points [tex]$(0, 10)$[/tex] and [tex]$(15, 0)$[/tex], and shade above this line (including the line).
### 2. The Maximum Number of Hours Lia Can Work at the Restaurant:
To find the maximum number of hours Lia can work at the restaurant ([tex]$r$[/tex]) while still meeting her earnings goal of at least [tex]$120$[/tex], we need to analyze the constraints:
- [tex]$r \geq 5$[/tex] means she must work at least 5 hours at the restaurant.
- [tex]$r + y \leq 15$[/tex] means she wants to work no more than 15 hours in total.
- [tex]$8r + 12y \geq 120$[/tex] represents her earnings goal constraint.
We need to find the feasible values of [tex]$r$[/tex] and [tex]$y$[/tex] that satisfy all three conditions.
Evaluating these constraints, we can isolate the condition involving earnings:
[tex]\[ 8r + 12y \geq 120 \][/tex]
First, express [tex]$y$[/tex] in terms of [tex]$r$[/tex] from [tex]$r + y \leq 15$[/tex]:
[tex]\[ y \leq 15 - r \][/tex]
Substituting [tex]$y = 15 - r$[/tex] into the earnings constraint:
[tex]\[ 8r + 12(15 - r) \geq 120 \][/tex]
[tex]\[ 8r + 180 - 12r \geq 120 \][/tex]
[tex]\[ -4r + 180 \geq 120 \][/tex]
[tex]\[ -4r \geq -60 \][/tex]
[tex]\[ r \leq 15 \][/tex]
Since [tex]$r$[/tex] must also be at least 5:
[tex]\[ 5 \leq r \leq 15 \][/tex]
Thus, the maximum number of hours Lia can work at the restaurant while still achieving her earnings goal is 15 hours.
### 3. The Maximum Amount of Money Lia Can Earn in 1 Week:
Given that Lia can work a maximum of 15 hours at the restaurant (from part 2), we will find her total earnings:
[tex]\[ y = 15 - 15 = 0 \][/tex]
Lia would work all 15 hours at the restaurant and none doing yard work:
Total earnings:
[tex]\[ 8r + 12y = 8(15) + 12(0) = 120 \][/tex]
Thus, the maximum amount of money Lia can earn in one week is [tex]$120. This aligns well with the constraints and ensures that she meets her earnings goal of at least $[/tex]120.
### Given Information:
- Lia must work at least 5 hours per week in the restaurant ([tex]$r \geq 5$[/tex]).
- She can work a maximum of 15 hours per week overall ([tex]$r + y \leq 15$[/tex]).
- She wants to earn at least [tex]$120 per week ($[/tex]8r + [tex]$12y \geq 120$[/tex]).
- Hourly wage at the restaurant is [tex]$8 per hour. - Hourly wage for yard work is $[/tex]12 per hour.
We'll answer each part of the question:
### 1. Graph the Inequalities:
To graph the inequalities:
- [tex]$r \geq 5$[/tex]:
Draw a vertical line at [tex]$r = 5$[/tex] and shade to the right (including the line).
- [tex]$r + y \leq 15$[/tex]:
Draw a line passing through points [tex]$(15, 0)$[/tex] and [tex]$(0, 15)$[/tex]. Shade below this line (including the line).
- [tex]$8r + 12y \geq 120$[/tex]:
Rearrange to [tex]$y \geq \frac{120 - 8r}{12} \Rightarrow y \geq 10 - \frac{2r}{3}$[/tex].
Draw this line by finding intercept points, let’s find the intercepts:
- Set [tex]$r = 0 \Rightarrow y = 10$[/tex]
- Set [tex]$y = 0 \Rightarrow $[/tex] [tex]$r = 15$[/tex]
Plot these points [tex]$(0, 10)$[/tex] and [tex]$(15, 0)$[/tex], and shade above this line (including the line).
### 2. The Maximum Number of Hours Lia Can Work at the Restaurant:
To find the maximum number of hours Lia can work at the restaurant ([tex]$r$[/tex]) while still meeting her earnings goal of at least [tex]$120$[/tex], we need to analyze the constraints:
- [tex]$r \geq 5$[/tex] means she must work at least 5 hours at the restaurant.
- [tex]$r + y \leq 15$[/tex] means she wants to work no more than 15 hours in total.
- [tex]$8r + 12y \geq 120$[/tex] represents her earnings goal constraint.
We need to find the feasible values of [tex]$r$[/tex] and [tex]$y$[/tex] that satisfy all three conditions.
Evaluating these constraints, we can isolate the condition involving earnings:
[tex]\[ 8r + 12y \geq 120 \][/tex]
First, express [tex]$y$[/tex] in terms of [tex]$r$[/tex] from [tex]$r + y \leq 15$[/tex]:
[tex]\[ y \leq 15 - r \][/tex]
Substituting [tex]$y = 15 - r$[/tex] into the earnings constraint:
[tex]\[ 8r + 12(15 - r) \geq 120 \][/tex]
[tex]\[ 8r + 180 - 12r \geq 120 \][/tex]
[tex]\[ -4r + 180 \geq 120 \][/tex]
[tex]\[ -4r \geq -60 \][/tex]
[tex]\[ r \leq 15 \][/tex]
Since [tex]$r$[/tex] must also be at least 5:
[tex]\[ 5 \leq r \leq 15 \][/tex]
Thus, the maximum number of hours Lia can work at the restaurant while still achieving her earnings goal is 15 hours.
### 3. The Maximum Amount of Money Lia Can Earn in 1 Week:
Given that Lia can work a maximum of 15 hours at the restaurant (from part 2), we will find her total earnings:
[tex]\[ y = 15 - 15 = 0 \][/tex]
Lia would work all 15 hours at the restaurant and none doing yard work:
Total earnings:
[tex]\[ 8r + 12y = 8(15) + 12(0) = 120 \][/tex]
Thus, the maximum amount of money Lia can earn in one week is [tex]$120. This aligns well with the constraints and ensures that she meets her earnings goal of at least $[/tex]120.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
