Question

Can someone please help me graph this...

Lia must work at least 5 hours per week in her familys restaurant for $8 per hour. She also does yard work for $12 per hour. Lias parents allow her to work a maximum of 15 hours per week overall. Lias goal is to earn at least $120 per week.

1.Write a system of inequalities to represent this situation. Let r be the number of hours worked at the restaurant, and let y be the number of hours of yard work.

2.Graph the inequalities.

Answer 1

Need Help Or Answers

Answer 2

This is as far as I've gotten \[5r+10y \le15\] and \[8r+12y \ge120\] but after that I am stuck.

Answer 3

Ill Explain

Answer 4

First identify the DECISION VARIABLES.
Here, the hours worked at the restaurant and at the yard are the two decision variables.
Let x= the number of hours worked at the restaurant and
y=the number of hours worked at the yard

Answer 5

Next step is to setup an OBJECTIVE FUNCTION/TARGET FUNCTION.
It must be something like Maximize/minimize
Here,profit is the Objective.
Total profit Lia can get if she work for $8 per hour in restaurant and $12 per hour in yard will be:
P=8x+12y
So our objective is to,
Maximize P=8x+12y

Answer 6

Next step is to form the CONSTRAINTS/RESTRICTIONS/REQUIREMENTS based on the availability of time and the earning.
Read the question from top to bottom then you will get those.

Answer 7

Now sketch the inequalities.This problem has only two decision variables.So it is the simplest Linear Programming Problem (LPP).Plot the inequalities on a 2-D graph.Note that an equality in LPP represents a line,so inequality represents a plane which is either to one side or the other side of the line.

Answer 8

I will explain how to select the plane for the inequality.
First plot the lines x=5, x+y=15, 8x+12y=120, x,y=0

Answer 9

Then take the vertical line x=5
x≥5 means plane right of x=5

Answer 10

For x+y=15
Lets put the origin (0,0) into above equation
0+0=0<15
That means the side for which the origin lies satisfy x+y≤15
So the region x+y≤15 is below the line x+y=15 (as shown in the graph)

Answer 11

Similarly for the line 8x+12y=120,
we can put (0,0) in the equation.
0+0<120
So the side where origin lie should satisfy 8x+12y≤120
But our required region 8x+12y≥120 is the side away from the origin (Above the line 8x+12y=120 as shown in the figure)

x≥0 means the region towards the right of y axis
y≥0 means the region above x axis

Answer 12

Then find the FEASIBLE REGION/SOLUTION SPACE.This region in the plane should SATISFY ALL THE CONSTRAINTS (including the non-negative restrictions).That means it is a common region for all the constraints.

Answer 13

Solution space can be Bounded(convex polygon) or Unbounded.
Note that the Optimum solution(or solutions) which maximizes or minimizes the objective function lies somewhere in the solution space.

Answer 14

Bounded problem is the simplest one.Fortunately this problem has a bounded solution space (As shown SHADED in the figure-ABC)

Answer 15

@Ellie202000 Are You Unhderstanding

Answer 16

@Ellie202000
your second equation is ok
the first one needs to be tweaked. you only want hours so you should not be multiplying r or y by any numbers

Answer 17

Ya I am starting to understand.

Answer 18

in other words, the sum of the hours worked at the restaurant and at yard work is \( \le 15\)

Answer 19

also, you should add one more inequality, which you get from
Lia must work at least 5 hours per week in her familys restaurant

Answer 20

Are You Good

Answer 21

So yard work is \[\ge15\] right?

Answer 22

Yes

Answer 23

@phi

Answer 24

the bigger side goes next to the "fat" end of \( \ge \)
the 15 (we can work upto 15 but not more) is the bigger number
the hours worked are r and y. added together r+y
try again

Answer 25

Yard work is \[\le 15\]

Answer 26

she is allowed to work 5 hours or 10 hours
but not 20 hours
r+y is less than or equal to 15

Answer 27

yes, but use r+y for the words yard work

Answer 28

allowed to work a maximum of 15 hours per week overall
means the hours at the restaurant (called r for short)
plus
hours at yard work (called y for short)
together must be less than or equal to 15

Answer 29

Ok so
r+y \[\le 15\]
y is\[\le10\]
r is\[\ge5\]

Right? I am thinking this because She has to work a minimum of five hour at her families restaurant so that mean she can work a maximum of ten hour doing yard work because her parents a allow her to work a maximum of 15 hours per week.

Answer 30

yes
\[ y\le 10\]
but we probably don't have to mention it.
(because as you said, that info comes from the other two relations)

now to plot them.
label the x axis "r" (in other words we will treat r as the x variable)
and let y (yard work) be the y-axis

do you know how to plot
\[ r \ge 5 \]
which we change (for this problem) into
\[ x \ge 5\]
?

first step: make it an equation:
x=5
do you know how to plot x=5 ?