Twenty first century electronics has discovered a theft problem at its warehouse and 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.
# of security guards: 1 1 2 3 4 5
# of radios stolen per week: 50 30 20 14 8 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 he 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 firm hire?
a 56m b 76 c 90
To determine the optimal number of security guards, we need to consider the trade-off between the cost of hiring security guards and the cost of stolen radios.
A. To calculate the total cost for each scenario, we'll multiply the number of security guards by their weekly pay rate ($200) and add the cost of stolen radios. Let's create a table to calculate the total cost for each scenario:
# of security guards | # of radios stolen per week | Total cost (in $)
------------------------------------------------------------------------
1 | 50 | ?
1 | 30 | ?
2 | 20 | ?
3 | 14 | ?
4 | 8 | ?
5 | 6 | ?
Now, let's calculate the total cost for each scenario:
For 1 security guard and 50 radios stolen per week:
Total cost = (1 * $200) + (50 * $25) = $200 + $1250 = $1450
For 1 security guard and 30 radios stolen per week:
Total cost = (1 * $200) + (30 * $25) = $200 + $750 = $950
For 2 security guards and 20 radios stolen per week:
Total cost = (2 * $200) + (20 * $25) = $400 + $500 = $900
For 3 security guards and 14 radios stolen per week:
Total cost = (3 * $200) + (14 * $25) = $600 + $350 = $950
For 4 security guards and 8 radios stolen per week:
Total cost = (4 * $200) + (8 * $25) = $800 + $200 = $1000
For 5 security guards and 6 radios stolen per week:
Total cost = (5 * $200) + (6 * $25) = $1000 + $150 = $1150
Based on the calculations, we can see that the lowest total cost is $900, which occurs when there are 2 security guards and 20 radios stolen per week. Therefore, the firm should hire 2 security guards.
B. To find out the most the firm would be willing to pay to hire the first security guard, we need to consider the scenario with 0 security guards. This scenario has the highest cost of stolen radios, which is 50 per week.
Total cost for 0 security guards:
Total cost = (0 * $200) + (50 * $25) = $0 + $1250 = $1250
Therefore, the most the firm would be willing to pay to hire the first security guard is $1250.
C. If the cost of a stolen radio increases to $50, we need to recalculate the total cost for each scenario, considering the higher cost.
For 1 security guard and 50 radios stolen per week:
Total cost = (1 * $200) + (50 * $50) = $200 + $2500 = $2700
For 1 security guard and 30 radios stolen per week:
Total cost = (1 * $200) + (30 * $50) = $200 + $1500 = $1700
For 2 security guards and 20 radios stolen per week:
Total cost = (2 * $200) + (20 * $50) = $400 + $1000 = $1400
For 3 security guards and 14 radios stolen per week:
Total cost = (3 * $200) + (14 * $50) = $600 + $700 = $1300
For 4 security guards and 8 radios stolen per week:
Total cost = (4 * $200) + (8 * $50) = $800 + $400 = $1200
For 5 security guards and 6 radios stolen per week:
Total cost = (5 * $200) + (6 * $50) = $1000 + $300 = $1300
Based on the calculations, we can see that the lowest total cost is $1200, which occurs when there are 4 security guards and 8 radios stolen per week. Therefore, the firm should hire 4 security guards.