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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1822.0. "Another word problem :^(" by RHINOS::OVEN () Wed Dec 08 1993 11:09

    	Suppose a company determines the cost of making a counterfeit
    game is
    		Cost(n,r,t) = 200 + nt + t*r^2
    
    			n = # of games
    			t = bribe money to be paid to the retailer
    			r = a factor that includes other expenses and
    			    costs.
    
    			n,r,t are always positive.
    
    	Suppose the bribe money increases by 50% and the r-factor
    increases by 10% with no increase in the number of games produced.
    What is the greatest percentage increase in costs the company
    can expect?
    
    			Any help would be great!
    
    					Thanks, Dave
T.RTitleUserPersonal
Name		DateLines
1822.1Consider terms separately.WIBBIN::NOYCEDEC 21064-200DX5 : 130 SPECint @ $36KWed Dec 08 1993 14:1111
80%.

Suppose the original t*r^2 is so large that the other terms are negligible.
Then the total increase in costs is 1.5*1.1*1.1 = 1.815.

This is the worst case -- if one of the other terms is large enough not to
be negligible, its increase will average the total down.  The n*t term only
increases by 1.5x, and the 200 term increases (?) by 1.0x.

Was this what you wanted, or did you want something in terms of the original
n, r, t?
1822.2RTL::GILBERTWed Dec 08 1993 14:3431
    The percentage increase is
    
    		Cost(n,1.1*r,1.5*t) - Cost(n,r,t)
    		--------------------------------- * 100
	    		    Cost(n,r,t)
    which expands to
    
    		0.5 * n*t + 0.815 * t*r^2
    		------------------------- * 100
		  200 + n*t + t*r^2
    
    		  0.5 * n + 0.815 * r^2
    		= --------------------- * 100 = F(n,r,t)
		    200/t + n + r^2
    
    Maximize this, subject to n,r,t all positive.
    
    First, any increase in t will increase F, because it makes the denominator
    smaller.  Any increase in r increases F, because it increases the
    numerator more than the denominator.  And any _decrease_ in n increases F,
    because the denominator is decreased more than then numerator.
    
    So
    	n -> 0
    	r -> infinity
    	t -> infinity
    And
    	F -> 0.815 * 100 = 100 * ( (1.5)*(1.1)^2 - 1 )
    
    So the maximum increase is 81.5%