Your furniture store sells two types of dining room tables. The first, type A, costs $265 and you make a $25 profit on each one. The second, type B, costs $100 and you make a $13 profit on each one. You can order no more than 40 tables this month, and you need to make at least $760 profit on them. If you must order at least one of each type of table, how many of each type of table should you order if you want to minimize your cost?

Question

Your furniture store sells two types of dining room tables. The first, type A, costs $265 and you make a $25 profit on each one. The second, type B, costs $100 and you make a $13 profit on each one. You can order no more than 40 tables this month, and you need to make at least $760 profit on them. If you must order at least one of each type of table, how many of each type of table should you order if you want to minimize your cost?
(1 point)
Responses

20 of type A; 20 of type B
20 of type A; 20 of type B

2 of type A; 38 of type B
2 of type A; 38 of type B

30 of type A; and 10 of type B
30 of type A; and 10 of type B

38 of type A; 2 of type B

GPT 3.5 · Answer

To minimize the cost, we need to find the combination that meets the profit requirement while minimizing the cost. Let's assume we order x tables of type A and y tables of type B.

The profit from type A tables is $25 per table, so the total profit from type A tables is 25x.
The profit from type B tables is $13 per table, so the total profit from type B tables is 13y.
According to the problem, the total profit should be at least $760, so we have the inequality: 25x + 13y ≥ 760.

We are also limited to ordering no more than 40 tables, so the total number of tables should be less than or equal to 40: x + y ≤ 40.

To find the combination that minimizes the cost, we can consider the cost of each type of table. The cost of type A tables is $265 per table, so the total cost of type A tables is 265x. The cost of type B tables is $100 per table, so the total cost of type B tables is 100y.

Now, let's solve the problem using linear programming:

Objective function:
Minimize the cost: Cost = 265x + 100y

Constraints:
25x + 13y ≥ 760
x + y ≤ 40
x ≥ 1 (we must order at least one of each type)

Based on the given options, let's check each possibility:

1. 20 of type A; 20 of type B:
Profit: 20 * 25 + 20 * 13 = $900
Cost: 20 * 265 + 20 * 100 = $7500
Not meeting the profit requirement.

2. 2 of type A; 38 of type B:
Profit: 2 * 25 + 38 * 13 = $614
Cost: 2 * 265 + 38 * 100 = $4730
Not meeting the profit requirement.

3. 30 of type A; 10 of type B:
Profit: 30 * 25 + 10 * 13 = $835
Cost: 30 * 265 + 10 * 100 = $9050
Not meeting the profit requirement.

4. 38 of type A; 2 of type B:
Profit: 38 * 25 + 2 * 13 = $1024
Cost: 38 * 265 + 2 * 100 = $10570
Profits meet the requirement.

Therefore, the optimal order would be 38 of type A and 2 of type B.