Party. We're told that student earns $10 per hour for tutoring in $7 per hour. As a teacher's aide, we're also told that X is number of hours spent tutoring and why is the number of hours per week as ah, teacher's aide, which are just abbreviated as a t. A. We were asked to write the objective function that models total weekly earnings. So it's called Z the total earnings per week. Then we have that SI is equal to well, student earns $10 per hour for tutoring, so there's $10 per hour times X hours per week is 10 x dollars per week tutoring plus and then a student earns $7 per hour as a ta. So seven why dollars per week as a t A. And so there in 10 X plus seven y dollars per week in total in Part B were given the constraints for the student and were asked to write a system of three inequalities modeling these constraints. So we're told that to have enough time for studies student can work no more than 20 hours per week, so we have the total number of hours per week which is X plus Y, must be less than or equal to 20. Tutoring center requires that each tour spend at least three hours per week tutoring. So we have that the number of hours in the tutoring center, which is Knicks, be at least three per week, and tutoring center also requires each tutor spend no more than eight hours per week tutoring. So we're also told that X is less than or equal to eight hours per week. Part C. You're asked to graft the system of inequalities in Part B. We're also told toe only use the first quadrant and its boundary. So before we graph it, let's just rationalize why we're only using the first quadrant. Well, this is because in addition to the three inequality is we found in Part B. We also have that since these air times, it must be greater than or equal to zero so X is greater than or equal to zero, and why has to be greater than or equal to zero? So, in part C, we're dealing only in the first Quadrant. And to graph the first inequality values red first, let's graph the equation X plus y equals 20. This has an X intercept of 20 0 in a Y intercept of 0 20 and since the line is a solution, I'll click these points with a solid line. I'll use the origin is a test point we have that zero is less than or equal to 20 so everything to the left of the red line is a solution. Autographed the second equation in green. Second inequality in green. So first graf X equals three. This is a vertical line at X equals three, and because the wine is a solution, I'll draw a solid line here, and everything to the right of this line is a solution, since the origin is not a solution and I'll draw the third inequality and blue do this autograph X equals eight. This is a vertical line at X equals eight, and since it's a solution, I'll draw a solid line, and we have that. Since the origin is a solution, everything to the left of this blue line is a solution, and so al draw the solution to the system of inequalities in black. The solution is the Quadrangle in Black, which has Vergis ease. 30 80 and then to find with these other two burgesses arm. Let's find the solutions to these systems of equations. So we have. They're section of the red and green lines, this is X plus y equals 20 and X equals three by simply substituting. We have that why is equal to 20 minus three, which is 17. And so we also have the Vertex 3 17 and it's last for Texas, the intersection of the red and blue lines. So this is X plus y equals 20 and X equals eight. Solve this system. Simply subtract he bottom equation from the top and we get why is equal to 12. So we have that eat 12 is the other vertex and parte de you're asked to evaluate the objective function at each of the vergis ease. So we have that or text. 30 Z is equal to 30. At Vertex 80 z is equal to 80. At Vertex 8 12 z is equal to well, this is going to be 80 plus 84 which is 1 64 And at Protects 3 17 z is equal to 30 plus and then 17 times seven. Just 70 plus 49 which is 119. So in 19 plus 30 which is 1 49 in Part E, we were asked to complete the missing portions of a statement. So, statement says the student can earn the maximum amount per week by tutoring for blank hours per week and working as a teacher's aide for blank hours per week. Maximum amount of student can earn each week is blank. So for the first blank, how many hours per week should it student tour to earn the maximum out? Well, recall that we have by a serum that if the objective function has a maximum over the region, determined by the system of linear inequalities, then the maximum will occur at at least one of the vergis ease of this region. And so, looking at her data from Part D, we have the maximum value of Z is 1 64 which occurs. That's the point 8 12 So we have that persons should spend okay hours per week tutoring and how many hours working as a teacher's aide 12 hours per week as a t A and the maximum amount of student can earn each week we determined WAAS 164 dollars per week