Jack and Jill are business partners who sell video game systems. The profit on Xbox One systems is $100 and the profit on PS4 systems is $80. Jack and Jill would like to sell at least 8 Xbox Ones and 10 PS4s but they were only shipped 20 systems this week. How many of each video game system must Jack and Jill sell in order to maximize profit for the week?

Jack and Jill must sell 10 systems each of PS4 and X-box One

Step-by-step explanation:

Solution:-

- The profit on Xbox One system, p1 = $100

- The profit on PS4 system, p2 = $80

- The number of systems shipped this week, N = 20

- The number of PS4 systems sold, n2 ≥ 10

- The number of Xbox One systems sold, n1 ≥ 8

- We are to maximize the total profit (P) for jack jill. The total profit made by Jack and jill will depend on the number of each system sold multiplied with their respective profits per system.

- We will mathematically express the total profit:

P = n1*p1 + n2*p2 ... Eq 1

Where,

- The total number of systems shipped are (N):

N = n1 + n2

20 = n1 + n2

- Develop an expression relating n1 and n2:

n1 = 20 - n2 ... Eq2

- Substitute Eq2 into Eq1:

P = (20 - n2) * 100 + n2*80

P = 2000 - 100*n2 + 80*n2

- To maximize the total profit (P) we need to set the first derivative of P with respect to number of PS4 system sold (n2) to zero and determine the maximizing value of (n2):

P = 2000 - 20*n2

- The range of number of PS4 systems to be sold is n2 ≥ 10. But from the profit function we see that number of PS4 systems sold must be small as possible. Hence, the lowest value for n2 = 10.

Hence,

- The number of X-box systems sold n1 would be:

n1 = 20 - 10

n1 = 10

Answer: Jack and Jill must sell 10 systems each of PS4 and X-box One.