Question 1195573
let x = the number of hours playing racquetball.
let y = the number of hours cycling.
your inequalities are:
x + y <= 12
this represents the total number of hours she wants to use is less than 12.
700x + 350y <= 7000
this represents the total number of calories she wants to burn.
it's 700 calories per hour times the number of hours for racquetball plus 350 calories per hour times the number of hours for cycling.
x >= 0
y >= 0
this represents that the number of hours for each activity can't be less than 0.
if she wants to minimize costs, then the objective function for costs is:
20x + 0y, which you want to minimize.
this represents 20 dollars per hour for racquetball plus 0 dollars for cycling.
if she wants maximize calories consumed, then the objective function for calories is:
700x + 350y, which you want to maximize.
this represents 700 calories per hour times number of hours for racquetball plus 350 calories per hour times number of hours for cycling.
you can graph this manually, or you can use the calculator at desmos.com.
if you use the calculator at desmos.com, you would:
graph the opposite of the inequalities.
the area of the graph that is not shaded will be your region of feasibility.
the corner points of the region of feasibility will be where your maximum calory burn and your minimum cost will be.
objective function for cost is 20x + 0y
objective function for calories is 700x + 350y
constraint inequalities are:
x + y <= 12
700x + 350y <= 7000
x >= 0
y >= 0
graph  the opposite of the inequalities.
identify the corner points.
evaluate the objective functions at the corner points.
the graph looks like this.
<img src = "http://theo.x10hosting.com/2022/072102.jpg">
the corner points of the feasible region are:
(0,12)
(8,4)
(10,0)
the minimum cost will be at (0,12), where the cost is 20 * 0 + 0 * 12 = 0.
the maximum calorie burn will be at (10,0) or (8,4).
at (10,0), the maximum calorie burn is 10 * 700 = 7000
at (8,4), the maximum calories burn is 8 * 700 + 4 * 350 = 7000.
you have a minimum cost of 0 and a maximum calorie burn of 7000.
i have no idea what the equation of 4x + 3y = 12 represents.
i don't see it as being applicable to this problem, unless there is some other information about the problem that you didn't show.
if you were to manually create the graph, you would do the following.
graph the equations of:
x + y = 12
700x + 350y = 7000
x = 0
y = 0
you would then shade the areas for the inequalities of:
x + y <= 12
700x + 3506 <= 7000
x >= 0
y >= 0
that graph would look like this:
<img src = "http://theo.x10hosting.com/2022/072103.jpg" >
using desmos.com calculator, the region of feasibility is the area that is  not shaded.
doing it manually, the region of feasibility is the area that is shaded.
the region of feasibility is the same, whichever way you wanted to graph it.
the corner points are the same and the analysis is the same.
note that, when she minimizes cost, the maximum calorie burn is 12 * 350 = 4200.
this is acceptable since it is less than 7000, which was one of the constraints.
all of the constraints need to be satisfied, not just some of them.
at (0,12), she had exercised for 12 hours which is <= 12 and she had burned 4200 calories which is <= 7000.
at (8,4), she had exercised for 12 hours which is <= 12 and she had burned 7000 calories which is <= 7000.
at (10,0), she had exercised for 12 hours which is <= 12 and she had burned 7000 calories which is <= 7000.
let me know if you have any questions.
theo