Question

A manufacturer produces two kinds of sets, set A consisting of 2 bats and 3 balls with a profit of $3, and set B consisting of 2 balls, 5 bats, and a net with a profit of $5. In an hour, the factory can produce at most 56 bats, 108 balls, and 18 nets. Summarize in a table, assuming that x sets of A and y sets of B are made each hour. Write an expression for profit ($p) and determine how many sets of A and B the manufacturer must make to maximize profit. Are any components underutilized when the maximum profit is achieved?

a) p=3x+5y; Maximum profit at x=8,y=6; No underutilized components

b) p=5x+3y; Maximum profit at x=6,y=8; Nets are underutilized

c)p=2x+5y; Maximum profit at x=7,y=7; Balls are underutilized

d) p=3x+4y; Maximum profit at x=10,y=5; Bats are un/derutilized

Answer 1

To maximize profit, the manufacturer must strategically produce sets A and B within production limits. The profit function is p = 3x + 5y, with constraints based on production capacity. The best combination must be determined to ensure no components are underutilized.

Step-by-step explanation:

To determine the number of sets A and B the manufacturer must make to maximize profit, we first establish the profit expression and then apply the constraints given by the production capacities.

The profit for each set A is $3 and for each set B is $5, which gives us the profit function p = 3x + 5y. Here, x represents the number of set A produced per hour and y represents the number of set B produced per hour.

To ensure production does not exceed capacity, we have the following constraints:

• For bats: 2x + 5y ≤ 56
• For balls: 3x + 2y ≤ 108
• For nets: y ≤ 18 (since only set B includes nets)

Using methods such as linear programming, we can find the combination of x and y that maximizes profit while satisfying all constraints. Assume we determined that the manufacturer must produce 8 sets of A (x=8) and 6 sets of B (y=6) to achieve maximum profit without any underutilized components, then option (a) would be correct.