An appliance store manager is ordering chest and upright freezers. One chest freezer costs $250 and delivers a $40 profit. One upright freezer costs $400 and delivers a $60 profit. Based on previous sales, the manager expects to sell at least 100 freezers. Total prfit must be at least $4800. Find the least number of each type of freezer the manager should order to minimize costs.

Let's assume the manager orders "x" chest freezers and "y" upright freezers.

Cost and Profit Equation:
Chest freezer cost: $250
Chest freezer profit: $40
Upright freezer cost: $400
Upright freezer profit: $60

The total cost equation is:
Total Cost = (Chest Freezer Cost * Number of Chest Freezers) + (Upright Freezer Cost * Number of Upright Freezers)
Total Cost = 250x + 400y

The total profit equation is:
Total Profit = (Chest Freezer Profit * Number of Chest Freezers) + (Upright Freezer Profit * Number of Upright Freezers)
Total Profit = 40x + 60y

Given Conditions:
Minimum Total Profit: $4800
Minimum Number of Freezers: 100

From the given conditions, we know that:
Total Profit >= $4800 ........(1)
Number of Chest Freezers + Number of Upright Freezers >= 100 ........(2)

Now, let's solve the equations to find the least number of each type of freezer the manager should order to minimize costs.

Step 1: Calculate the minimum profit condition
40x + 60y >= 4800 ........(1)

Step 2: Simplify the profit condition
2x + 3y >= 240 ........(3) [Divided by 20]

Step 3: Calculate the minimum number of freezers condition
x + y >= 100 ........(2)

Step 4: Graph the inequalities

Let's graph the lines (2x + 3y = 240) and (x + y = 100) to find the feasible region.

In the graph, the shaded region represents the feasible region satisfying both conditions.

Step 5: Find the vertices of the feasible region

The vertices of the feasible region are the points where the lines intersect.
In this case, we have three points: (20, 80), (40, 60), and (100, 0).

Step 6: Substitute the vertices into the total cost equation

Substituting the vertices into the total cost equation: Total Cost = 250x + 400y
For (20, 80):
Total Cost = (250*20) + (400*80) = $38,000
For (40, 60):
Total Cost = (250*40) + (400*60) = $38,000
For (100, 0):
Total Cost = (250*100) + (400*0) = $25,000

Step 7: Determine the minimum cost

Among the three calculated costs, the minimum cost is $25,000 when the manager orders 100 chest freezers (x = 100) and 0 upright freezers (y = 0).

Therefore, the manager should order 100 chest freezers and 0 upright freezers to minimize costs.

To find the least number of each type of freezer the manager should order to minimize costs, we need to set up equations using the given information and solve them.

Let's assume the manager orders x chest freezers and y upright freezers.

The cost of x chest freezers is $250x.
The cost of y upright freezers is $400y.

The profit from selling x chest freezers is $40x.
The profit from selling y upright freezers is $60y.

The total cost can be represented as the sum of the two costs:
Total Cost = $250x + $400y

The total profit can be represented as the sum of the two profits:
Total Profit = $40x + $60y

The constraints given are:
1. The manager expects to sell at least 100 freezers, so x + y ≥ 100. (Equation 1)
2. The total profit must be at least $4800, so $40x + $60y ≥ $4800. (Equation 2)

To minimize costs, we need to minimize the total cost. This can be achieved by minimizing the individual costs of chest and upright freezers.

To solve this problem, we can use the method of linear programming, specifically the Simplex method or graphical method. However, the details of solving linear programming problems are beyond the scope of this explanation.