Twenty-first century electronics has discovered a theft problem at its warehouse and has decided to hire security guards. The firm wants to hire the optimal number of security guards. The following table shows how the number of security guards affects the number of radios stolen per week.
Number of security guards: Number of radios stolen per week
0 50
1 30
2 20
3 14
4 8
5 6
A). If each security guard is paid $200 a week and the cost of a stolen radio is $ 25, how many security guards should the firm hire?
B). If the cost of a stolen radio is $ 25, what is the most the firm would be willing to pay to hire the first security guard?
C). If each security guard is paid $ 200 a week and the cost of a stolen radio is $50, how many security guards should the guard hire?
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To determine the optimal number of security guards to hire and answer the given questions, we need to analyze the data provided in the table.
A) To find the optimal number of security guards to hire, we need to minimize the total cost, which consists of the cost of security guards and the cost of stolen radios. We can calculate the total cost for each given number of security guards by multiplying the number of stolen radios by the cost per radio and adding it to the cost of security guards.
Let's calculate the total cost for each scenario:
Number of security guards: 0
Number of radios stolen per week: 50
Cost of stolen radios: $25
Cost of security guards: $200 x 0 = $0
Total cost: 50 x $25 + $0 = $1250
Number of security guards: 1
Number of radios stolen per week: 30
Cost of stolen radios: $25
Cost of security guards: $200 x 1 = $200
Total cost: 30 x $25 + $200 = $950
Number of security guards: 2
Number of radios stolen per week: 20
Cost of stolen radios: $25
Cost of security guards: $200 x 2 = $400
Total cost: 20 x $25 + $400 = $900
Number of security guards: 3
Number of radios stolen per week: 14
Cost of stolen radios: $25
Cost of security guards: $200 x 3 = $600
Total cost: 14 x $25 + $600 = $950
Number of security guards: 4
Number of radios stolen per week: 8
Cost of stolen radios: $25
Cost of security guards: $200 x 4 = $800
Total cost: 8 x $25 + $800 = $1000
Number of security guards: 5
Number of radios stolen per week: 6
Cost of stolen radios: $25
Cost of security guards: $200 x 5 = $1000
Total cost: 6 x $25 + $1000 = $1150
Based on the calculations, the firm should hire 2 security guards, as it results in the lowest total cost of $900.
B) To determine the most the firm would be willing to pay to hire the first security guard, we need to find the change in total cost when going from 0 to 1 security guard. By subtracting the total cost of 0 security guards from the total cost of 1 security guard, we can determine the maximum amount the firm is willing to pay.
Total cost with 0 security guards: $1250
Total cost with 1 security guard: $950
Change in total cost: $950 - $1250 = -$300
The firm would be willing to pay up to $300 to hire the first security guard.
C) If each security guard is paid $200 a week and the cost of a stolen radio is $50, we repeat the process from part A with the updated costs.
Number of security guards: 0
Total cost: 50 x $50 + $0 = $2500
Number of security guards: 1
Total cost: 30 x $50 + $200 = $1700
Number of security guards: 2
Total cost: 20 x $50 + $400 = $1400
Number of security guards: 3
Total cost: 14 x $50 + $600 = $1300
Number of security guards: 4
Total cost: 8 x $50 + $800 = $1200
Number of security guards: 5
Total cost: 6 x $50 + $1000 = $1300
Based on the calculations, the firm should hire 4 security guards, as it results in the lowest total cost of $1200 when considering the increased cost of stolen radios.
Note: These calculations assume a linear relationship between the number of security guards and the number of radios stolen, which may not be entirely accurate in a real-world scenario. The data provided in the table is limited, and other factors may need to be considered for a more comprehensive analysis.