A factory produces two products A and B using raw materials p and q. It uses 3 units of p and 1 unit of q to produce one unit of A. It uses 1 unit of p and 2 units of q to produce one unit of B. The maximum daily supplies are 330 units of p and 220 units of q. If the profit per unit of A is $3 and the profit per unit of B is $2, what is the maximum total daily profit (in dollars)?
If x units of A are made, and y units of B, then
3x+y <= 330
x+2y <= 220
p(x,y) = 3x+2y
plug into your favorite linear algebra calculator to find
maximum p(88,66) = 396
To find the maximum total daily profit, we need to determine how many units of products A and B can be produced within the given constraints, and then calculate the profit for each product.
Let's start by figuring out how many units of A and B can be produced using the available raw materials.
Given that each unit of A requires 3 units of p and 1 unit of q, we can calculate the maximum number of units of A that can be produced:
Max units of A = (available units of p) / (units of p per unit of A) = 330 / 3 = 110
Similarly, for product B, we have:
Max units of B = (available units of q) / (units of q per unit of B) = 220 / 2 = 110
Now, let's calculate the profit for each product.
Profit per unit of A = $3
Profit per unit of B = $2
So, the total daily profit for A is:
Total profit for A = (max units of A) * (profit per unit of A) = 110 * 3 = $330
And the total daily profit for B is:
Total profit for B = (max units of B) * (profit per unit of B) = 110 * 2 = $220
Finally, we can find the maximum total daily profit by summing up the profits from A and B:
Maximum total daily profit = Total profit for A + Total profit for B = $330 + $220 = $550
Therefore, the maximum total daily profit is $550.