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Please help set up a derivative problem for business calc.

Suppose the annual cost per active-duty armed service member in a certain country increased from $80,000 in 1995 to $90,000 in 2000. In 1990, there were 2 million armed service personnel and this number decreased to 1.5 million in 2000. Use linear models for annual cost and personnel to estimate, to the nearest $10 million, the rate of change of total military personnel costs in 1998.
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1 Answer

Let Cp= cost/active duty person in $
      N= Number on active duty in millions
Using linear models and the data provided we have
 
Cp(T)=(10,000/5)*(T-1995) + 80,000. (note that when T=1995 Cp(1995)=80,000 and when T=2000 Cp(2000)=90,000.)
 

N(T)= -.05*(T-1990) + 2 , (Note the rate of change here is <0 troop strength is decreasing over time)

 
The total cost CT(T)= Cp(T)*N(T) and the rate of change is dCT/dT=Cp*dN/dT + N*dCp/dT in millions $/yr
Calculate each term
 
dN/dT=-.05 and dCp/dT=2000 (note the rates of change are constant)
Cp(1998)=2000*3 + 80000=86,000
N(1998)=-.05*8 + 2=1.6
 
Putting all the terms together
dCT/dT=Cp*dN/dT + N*dCp/dT=86,000*(-.05)+1.6*2000 =-1,100 in millions $/yr
So the rate of change of total cost in 1998 was -1,100,000 $/yr
 
Jim