Two woodworkers, Tom and Carlos, bring in $100 for making a table and $80 for making a chair. On average, Tom must work 3 hours and Carlos 2 hours to make a chair. Tom must work 2 hours and Carlos 6 hours to make a table. If neither wishes to work more than 42 hours per week, how many tables and how many chairs should they make to maximize their income?

Question

Two woodworkers, Tom and Carlos, bring in $100 for making a table and $80 for making a chair. On average, Tom must work 3 hours and Carlos 2 hours to make a chair. Tom must work 2 hours and Carlos 6 hours to make a table. If neither wishes to work more than 42 hours per week, how many tables and how many chairs should they make to maximize their income?

Answer 1

For x tables and y chairs, we want to

maximize p=100x+80y subject to
3x+2y <= 42
2x+6y <= 42

Now use your favorite linear programming tool to find that they make the most with

12 tables and 3 chairs.

Answer 2

To maximize their income, Tom and Carlos should determine the number of tables and chairs they should make, considering their hourly rates and maximum working hours.

Let's assume Tom makes t tables and Carlos makes c chairs.

The total income for the tables made by Tom would be $100 multiplied by the number of tables (t). The total income for the chairs made by Carlos would be $80 multiplied by the number of chairs (c).

Therefore, the total income for Tom and Carlos would be:
Total Income = (100t) + (80c)

Now let's consider the working hours for Tom and Carlos:

Tom takes 2 hours to make a table, and Carlos takes 6 hours.
Tom takes 3 hours to make a chair, and Carlos takes 2 hours.

For Tom, the total working hours for tables would be 2t, and for chairs, it would be 3c.
For Carlos, the total working hours for tables would be 6t, and for chairs, it would be 2c.

Since the maximum working hours per week for each woodworker is 42 hours, we can set up the following equations:

2t + 3c ≤ 42 (Tom's maximum working hours)
6t + 2c ≤ 42 (Carlos's maximum working hours)

To further simplify these equations, we can divide each equation by 2:

t + (3/2)c ≤ 21 (Tom's maximum working hours)
3t + c ≤ 21 (Carlos's maximum working hours)

Now, we need to determine the feasible region by graphing these inequalities. However, since this is a step-by-step bot without a graphical functionality, we can use a logical approach to solve the problem.

Let's consider the maximum values of t and c that satisfy the inequalities:

For Tom's maximum working hours (t + (3/2)c ≤ 21):
1. If t = 0, then (3/2)c ≤ 21, which means c ≤ 14.
2. If c = 0, then t ≤ 21.
3. If c = 14, then t ≤ 21 - (3/2)(14), which reduces to t ≤ 0.

Therefore, for Tom's maximum working hours, the feasible values for t range from 0 to 0, and for c, it ranges from 0 to 14.

Now let's consider Carlos's maximum working hours (3t + c ≤ 21):
1. If t = 0, then c ≤ 21.
2. If c = 0, then 3t ≤ 21, which means t ≤ 7.
3. If t = 7, then c ≤ 21 - 3(7), which reduces to c ≤ 0.

Therefore, for Carlos's maximum working hours, the feasible values for t range from 0 to 7, and for c, it ranges from 0 to 0.

Now, we can evaluate the maximum income within the feasible region:

The total income = (100t) + (80c)

Substituting the values for the feasible region:
- When t = 0 and c = 0, the total income = (100 * 0) + (80 * 0) = 0.
- When t = 0 and c = 14, the total income = (100 * 0) + (80 * 14) = 1120.
- When t = 7 and c = 0, the total income = (100 * 7) + (80 * 0) = 700.

Therefore, they should make 7 tables and 0 chairs to maximize their income, resulting in a total income of $700.

Answer 3

To maximize their income, Tom and Carlos need to determine the number of tables and chairs they should make within the 42-hour limit. Let's denote the number of tables as 't' and the number of chairs as 'c'.

Since Tom takes 2 hours to make a table, he can work for a maximum of 42/2 = 21 hours on tables. Therefore, the maximum number of tables Tom can make is 21/2 = 10.5, but since we can't have a fraction of a table, Tom can make a maximum of 10 tables.

Similarly, since Tom takes 3 hours to make a chair, he can work for a maximum of 42/3 = 14 hours on chairs. Hence, the maximum number of chairs Tom can make is 14/2 = 7 chairs.

On the other hand, Carlos takes 6 hours to make a table, so he can work for a maximum of 42/6 = 7 hours on tables. Thus, the maximum number of tables Carlos can make is 7/6 ≈ 1.17, but since we can't make a fraction of a table, Carlos can make a maximum of 1 table.

Lastly, Carlos takes 2 hours to make a chair, so he can work for a maximum of 42/2 = 21 hours on chairs. Therefore, the maximum number of chairs Carlos can make is 21/2 = 10.5, but since we can't have a fraction of a chair, Carlos can make a maximum of 10 chairs.

Now, let's consider the different scenarios of table and chair combinations:

1. Tom makes 10 tables (earning $1000) and Carlos makes 10 chairs (earning $800), totaling $1800.
2. Tom makes 10 tables (earning $1000) and Carlos makes 9 chairs (earning $720), totaling $1720.
3. Tom makes 10 tables (earning $1000), and Carlos makes 8 chairs (earning $640), totaling $1640.
4. Tom makes 9 tables (earning $900), and Carlos makes 10 chairs (earning $800), totaling $1700.
5. Tom makes 8 tables (earning $800), and Carlos makes 10 chairs (earning $800), totaling $1600.

The combination that maximizes their income is scenario 1, where Tom makes 10 tables and Carlos makes 10 chairs. This results in a total income of $1800.