An appliance store manager is ordering chest and upright freezers. one chest freezers 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 profit must be at least $4800. Find the least number of each type of freezer the manager should order to minimize costs.
c = # of chest freezers
u = # of upright freezers
c+u>=100 or u >= 100-c
assume a total of 100 units
cost = 250c + 400u
= 250c +400(100-c)
= 250c + 40000 - 400c
= 40000 - 150c = 800 - 3c
so obviously the more type c units, the lower the total cost.
profit = 40c + 60u >= 4800
substitute u in terms of c in the profit equation to find the maximum number of type c units to maintain a minumum profit of 4800
To find the least number of each type of freezer the manager should order to minimize costs, we can set up a system of linear inequalities based on the given information.
Let's denote the number of chest freezers as 'C' and the number of upright freezers as 'U.'
Based on the cost and profit of each type of freezer, we can write the following equations:
Cost Equation: 250C + 400U ≤ Total Cost
Profit Equation: 40C + 60U ≥ 4800
The goal is to minimize costs, so we will focus on the cost equation.
Given that the manager expects to sell at least 100 freezers, we can add an additional constraint:
C + U ≥ 100
Now, we can solve this system of linear inequalities to find the least number of each type of freezer the manager should order.
First, let's solve the constraint equation:
C + U ≥ 100
We could rewrite this as:
C ≥ 100 - U
Now, substitute this inequality into the cost equation:
250C + 400U ≤ Total Cost
250(100 - U) + 400U ≤ Total Cost
25000 - 250U + 400U ≤ Total Cost
150U ≤ Total Cost - 25000
U ≤ (Total Cost - 25000) / 150
Therefore, the inequality for U is U ≤ (Total Cost - 25000) / 150.
Now, let's look at the profit equation:
40C + 60U ≥ 4800
We can rewrite this as:
C ≥ (4800 - 60U) / 40
Therefore, the inequality for C is C ≥ (4800 - 60U) / 40.
Now, we have the following system of inequalities:
C + U ≥ 100
C ≥ (4800 - 60U) / 40
U ≤ (Total Cost - 25000) / 150
To find the least number of each type of freezer, we need to minimize the cost. This can be achieved by simultaneously maximizing the number of upright freezers (U) and minimizing the number of chest freezers (C) within the given constraints.
We would need to find the values of U and C that satisfy all three inequalities while minimizing C.
Please note that the problem doesn't provide the value for the "Total Cost." To find the exact number of each type of freezer, we would need to know the total cost.