Question

A craftsman can sell 10 jewelry sets for $500 each. He knows that for each additional set he makes, the price of each set will decrease by $25. How many jewelry sets should he make if he wants to maximize his earnings?

Answer 1

To maximize his earnings, the craftsman should make an additional 10 jewelry sets.

Step-by-step explanation:

To maximize his earnings, the craftsman should continue making jewelry sets as long as the decrease in price does not exceed the increase in quantity. Let's calculate the optimal number of sets:

1. The initial price of each set is $500.

2. For each additional set made, the price decreases by $25.

3. Let's assume the craftsman makes x additional sets. The new price per set is:

$500 - $25x.

4. The total revenue from selling 10 sets at $500 each is:

$500 x 10 = $5000.

5. The revenue from selling x additional sets at $500 - $25x each is:

($500 - $25x) x x.

6. To maximize earnings, we need to maximize the revenue. So, the total revenue is:

$5000 + ($500 - $25x) x x.

7. We can expand the expression and simplify it to find the maximum:

$5000 + (500x - 25x^2).

8. To find the maximum, we can take the derivative of the expression and set it equal to 0:

d/dx ($5000 + (500x - 25x^2)) = 0.

9. Solving the equation, we find x = 10.

10. Therefore, the craftsman should make an additional 10 sets to maximize his earnings.

Answer 2

The max warning happens when he makes 15 sets.

Here is what happens as he makes more and more sets:

10 sets: $500 each
Total earned: 10*500=5000
11 sets: $475 each
Total earned: 11*475=5225
12 sets: $450 each
Total earned: 12*450=5400
13 sets: $425 each
Total earned: 13*425=5525
14 sets: $400 each
Total earned: 14*400=5600
15 sets: $375 each
Total earned: 15*375=5625
16 sets: $350 each
Total earned: 16*350=5600
17 sets: $325 each
Total earned: 17*325=5525
18 sets: $300 each
Total earned: 18*300=5400
19 sets: $275 each
Total earned: 19*275=5225
20 sets: $250 each
Total earned: 20*250=5000
21 sets: $225 each
Total earned: 21*225=4725
22 sets: $200 each
Total earned: 22*200=4400
23 sets: $175 each
Total earned: 23*175=4025
24 sets: $150 each
Total earned: 24*150=3600
25 sets: $125 each
Total earned: 25*125=3125
26 sets: $100 each
Total earned: 26*100=2600