21st 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 guards. The follwoign table shows how the number of security guards affects the number of radios stolen each week:
# of guards # of radios stolen per wk
0 50
1 30
2 20
3 14
4 8
5 6
a) If each security guards is paid
$200 a week and the cost of a
stolen radio is $25, how many
security guards shoudl 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 guards
should the firm hire?
Make a third column, called Cost. Cost will be the cost of the security guard and the value of the radios stolen. You want to minimize this.
G Stolen Cost
0 50 1250
1 30 950
2 20 900 and so on
To find the optimal number of security guards the firm should hire, we can calculate the cost for each scenario and choose the one with the minimum cost. Let's go through the calculations step by step:
a) To calculate the cost, we need to multiply the number of security guards by their weekly wage and add the value of the radios stolen. This will give us the total cost for each scenario:
G (Number of guards) / Stolen (Radios stolen per week) / Cost
0 / 50 / 0 guards * $200 = $0 + 50 radios * $25 = $1250
1 / 30 / 1 guard * $200 = $200 + 30 radios * $25 = $950
2 / 20 / 2 guards * $200 = $400 + 20 radios * $25 = $900
3 / 14 / 3 guards * $200 = $600 + 14 radios * $25 = $850
4 / 8 / 4 guards * $200 = $800 + 8 radios * $25 = $1000
5 / 6 / 5 guards * $200 = $1000 + 6 radios * $25 = $1150
From the table, we can see that the minimum cost is $850 when the firm hires 3 security guards.
b) To find the most the firm would be willing to pay to hire the first security guard, we need to calculate the difference in cost when going from 0 to 1 security guard:
Cost of first guard = Cost with 1 guard - Cost with 0 guards
= $950 - $1250
= -$300
Since the firm would save $300 by hiring the first guard, they could be willing to pay up to $300 for hiring the first security guard.
c) To find the number of guards the firm should hire when each security guard is paid $200 a week and the cost of a stolen radio is $50, we can follow the same steps in part a:
G (Number of guards) / Stolen (Radios stolen per week) / Cost
0 / 50 / 0 guards * $200 = $0 + 50 radios * $50 = $2500
1 / 30 / 1 guard * $200 = $200 + 30 radios * $50 = $1700
2 / 20 / 2 guards * $200 = $400 + 20 radios * $50 = $1400
3 / 14 / 3 guards * $200 = $600 + 14 radios * $50 = $1200
4 / 8 / 4 guards * $200 = $800 + 8 radios * $50 = $1000
5 / 6 / 5 guards * $200 = $1000 + 6 radios * $50 = $1300
From the table, we can see that the minimum cost is $1000 when the firm hires 4 security guards.