A student earns $10 per hour for tutoring and $7 per hour as a teacher's aide. To have enough free time for studies, he can work no more than 20 hours per week. The tutoring center requires that each tutor spends at least three hours per week tutoring, but no more than eight hours per week. How many hours should he work to maximize his earnings?
hours of tutoring
hours as a teacher's aide
What is the maximum profit? $
164 at (8,12)
http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=1e692c6f72587b2cbd3e7be018fd8960&title=Linear%20Programming%20Calculator&theme=blue
max
10x+7y
x+y<=20
x>=3
x<=8
x+y>=1 phony to fill all the blanks
did with linear programming, no need for simplex
To determine the maximum earnings, we need to find the optimal number of hours for tutoring and being a teacher's aide. Let's break down the constraints and calculate the earnings for each scenario.
1. Tutoring hours:
- Minimum tutoring hours required: 3 hours/week
- Maximum tutoring hours allowed: 8 hours/week
- Earnings per hour: $10
2. Teacher's aide hours:
- Total working hours (including tutoring): 20 hours/week
- Teacher's aide hours = Total working hours - Tutoring hours
Now, let's calculate the earnings for different scenarios and find the combination that maximizes the total earnings.
Case 1: When the student tutors for 3 hours and spends the remaining time as a teacher's aide.
- Tutoring hours: 3 hours
- Teacher's aide hours: 20 - 3 = 17 hours
- Total earnings: (3 hours x $10) + (17 hours x $7) = $97
Case 2: When the student tutors for 4 hours and spends the remaining time as a teacher's aide.
- Tutoring hours: 4 hours
- Teacher's aide hours: 20 - 4 = 16 hours
- Total earnings: (4 hours x $10) + (16 hours x $7) = $108
Continue calculating the earnings for different tutoring hours within the given constraints until we find the maximum.
Case 3: When the student tutors for 5 hours and spends the remaining time as a teacher's aide.
- Tutoring hours: 5 hours
- Teacher's aide hours: 20 - 5 = 15 hours
- Total earnings: (5 hours x $10) + (15 hours x $7) = $115
Case 4: When the student tutors for 6 hours and spends the remaining time as a teacher's aide.
- Tutoring hours: 6 hours
- Teacher's aide hours: 20 - 6 = 14 hours
- Total earnings: (6 hours x $10) + (14 hours x $7) = $122
Case 5: When the student tutors for 7 hours and spends the remaining time as a teacher's aide.
- Tutoring hours: 7 hours
- Teacher's aide hours: 20 - 7 = 13 hours
- Total earnings: (7 hours x $10) + (13 hours x $7) = $127
Case 6: When the student tutors for 8 hours and spends no time as a teacher's aide.
- Tutoring hours: 8 hours
- Teacher's aide hours: 20 - 8 = 12 hours
- Total earnings: (8 hours x $10) + (12 hours x $7) = $136
From the above calculations, we can see that the maximum earnings of $136 can be achieved when the student works 8 hours as a tutor and doesn't work as a teacher's aide.
Therefore, the answer to the first question is:
- Hours of tutoring = 8
To calculate the hours as a teacher's aide:
- Tutoring center requires at least 3 hours of tutoring, so the remaining hours will be allocated to being a teacher's aide.
- Hours as a teacher's aide = 20 - 8 = 12
Therefore, the answer to the second question is:
- Hours as a teacher's aide = 12
To find the number of hours he should work to maximize his earnings, we need to consider two factors: the number of hours he spends tutoring and the number of hours he spends as a teacher's aide.
Let's denote the number of hours he spends tutoring as 't' and the number of hours he spends as a teacher's aide as 'a'. Since the tutoring center requires each tutor to spend at least three hours per week tutoring and the student can work a maximum of 20 hours per week, this tells us that:
3 ≤ t ≤ 8 (tutoring hours)
Also, since he can work a maximum of 20 hours per week, we have the following constraint:
t + a ≤ 20 (total hours)
Now let's calculate his earnings. He earns $10 per hour for tutoring and $7 per hour as a teacher's aide.
His earnings from tutoring would be 10 * t and his earnings from being a teacher's aide would be 7 * a.
To maximize his earnings, we need to maximize the total earnings function:
E(t, a) = 10t + 7a
To solve this problem, we can use a technique known as linear programming or graphical methods. However, since there are only a few possible values for t and a in this scenario, we can simply evaluate the earnings for each possible combination and determine the maximum.
Let's consider the possible combinations of t and a:
1) t = 3, a = 0
2) t = 4, a = 0
3) t = 5, a = 0
4) t = 6, a = 0
5) t = 7, a = 0
6) t = 8, a = 0
7) t = 2, a = 1
8) t = 3, a = 1
9) t = 4, a = 1
10) t = 5, a = 1
11) t = 6, a = 1
12) t = 7, a = 1
13) t = 2, a = 2
14) t = 3, a = 2
15) t = 4, a = 2
16) t = 5, a = 2
17) t = 6, a = 2
18) t = 2, a = 3
19) t = 3, a = 3
20) t = 4, a = 3
21) t = 5, a = 3
22) t = 2, a = 4
23) t = 3, a = 4
24) t = 4, a = 4
25) t = 2, a = 5
26) t = 3, a = 5
27) t = 2, a = 6
Now, we can calculate the earnings for each combination:
1) E(3, 0) = (10 * 3) + (7 * 0) = 30
2) E(4, 0) = (10 * 4) + (7 * 0) = 40
3) E(5, 0) = (10 * 5) + (7 * 0) = 50
4) E(6, 0) = (10 * 6) + (7 * 0) = 60
5) E(7, 0) = (10 * 7) + (7 * 0) = 70
6) E(8, 0) = (10 * 8) + (7 * 0) = 80
7) E(2, 1) = (10 * 2) + (7 * 1) = 24
8) E(3, 1) = (10 * 3) + (7 * 1) = 37
9) E(4, 1) = (10 * 4) + (7 * 1) = 47
10) E(5, 1) = (10 * 5) + (7 * 1) = 57
11) E(6, 1) = (10 * 6) + (7 * 1) = 67
12) E(7, 1) = (10 * 7) + (7 * 1) = 77
13) E(2, 2) = (10 * 2) + (7 * 2) = 31
14) E(3, 2) = (10 * 3) + (7 * 2) = 44
15) E(4, 2) = (10 * 4) + (7 * 2) = 54
16) E(5, 2) = (10 * 5) + (7 * 2) = 64
17) E(6, 2) = (10 * 6) + (7 * 2) = 74
18) E(2, 3) = (10 * 2) + (7 * 3) = 38
19) E(3, 3) = (10 * 3) + (7 * 3) = 51
20) E(4, 3) = (10 * 4) + (7 * 3) = 61
21) E(5, 3) = (10 * 5) + (7 * 3) = 71
22) E(2, 4) = (10 * 2) + (7 * 4) = 45
23) E(3, 4) = (10 * 3) + (7 * 4) = 58
24) E(4, 4) = (10 * 4) + (7 * 4) = 68
25) E(2, 5) = (10 * 2) + (7 * 5) = 52
26) E(3, 5) = (10 * 3) + (7 * 5) = 65
27) E(2, 6) = (10 * 2) + (7 * 6) = 59
Now we can compare the earnings for the different combinations and determine the maximum earnings and the corresponding values for t and a.
The maximum earnings among all these combinations are $80, which occurs when the student works 8 hours as a tutor and 0 hours as a teacher's aide. Therefore, he should work 8 hours as a tutor to maximize his earnings.
To summarize:
Hours of tutoring = 8
Hours as a teacher's aide = 0
Maximum profit = $80