A manufacturer of projection TVs must ship a total of at least 1000 TVs to its two central warehouses, x to the first warehouse and y to the second warehouse. Each warehouse can hold a maximum of 750 TVs. The first warehouse already has 150 TVs on hand, whereas the second has 50 TVs on hand. It costs $9 to ship a TV to the first warehouse, and it costs $16 to ship a TV to the second warehouse. How many TVs should be shipped to each warehouse to minimize cost?
(x, y) =
What is the minimum cost?
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$9 (750-150) = $9 * 600 = ?
$16(1000-600) = ?
Add the two ? to get your answer.
To minimize the cost, we need to find the number of TVs that should be shipped to each warehouse (x and y) in order to satisfy the given conditions. We can approach this problem using linear programming.
First, let's define our variables:
x = number of TVs to be shipped to the first warehouse
y = number of TVs to be shipped to the second warehouse
Next, let's establish the constraints:
1. The total number of TVs shipped should be at least 1000: x + y >= 1000
2. The first warehouse can hold a maximum of 750 TVs: x <= 750 - 150 (subtract the TVs already on hand)
3. The second warehouse can hold a maximum of 750 TVs: y <= 750 - 50 (subtract the TVs already on hand)
Now, let's define the objective function, which is the cost:
Cost = 9x + 16y
We need to minimize this cost.
To solve this linear programming problem, we need to plot the constraints on a graph and find the feasible region. However, we can simplify the problem by considering the extremes of the constraints.
Let's analyze the variables and constraints:
1. Since we need at least 1000 TVs in total, let's consider the case when x = 1000 and y = 0 to satisfy this minimum requirement.
Cost = 9(1000) + 16(0) = 9000
2. Now let's consider the case when x = 0 and y = 1000 to satisfy the minimum requirement.
Cost = 9(0) + 16(1000) = 16000
From the above analysis, we can see that the minimum cost will be achieved either when x = 1000 and y = 0 or when x = 0 and y = 1000.
So, (x, y) can be one of the following pairs:
(1000, 0)
(0, 1000)
In both cases, the minimum cost is $9000.
Therefore, (x, y) = (1000, 0) with a minimum cost of $9000.