MAXIMA AND MINIMA 125 



The dimensions of the conical vessel can now be given. 



Radius - 



Height =yR 2 -gR 2 = VJ R 



TC 2R 2 /I 

 Volume = A/- R 



27CR 3 



" 9\/3 



78. Example 6. The cost of a ship per hour is c where 

 c = a + bs n and a, b, and n are constants and * is the speed in 

 knots. Find the speed of the ship so that it will travel a passage 

 of m nautical miles at a minimum total cost. 



Tfl 



Time of passage = hours 



Total cost of passage = (a + bs n ) pounds = y 



S 



y = m(^+bs< 



v will be a minimum when ~ = 



ds 



< jj- = m{ - as- 2 + b(n - l)s n ~ 2 } 



and -r = when b(n l)s n ~ 2 = % 



ds s 2 



s n = 



b(n - 1) 



r_o_\5 

 = U(n-l)/ 



giving the speed in terms of the known constants a, b, and n. 



79. The function t/=oe~ w sin(ptc) affords a striking example 

 of alternating maximum and minimum values. 



a, &, p, and c are constants. 



Since e-** is never zero, y = when sin (pt c) = 0. That is, 

 when (pt c} has the values 0, TT, 27T, STC . . . nit . . . or when 



C C n C 27T C 717C 



/ has the values -, - H , - H ...-H ... 



p p p p p P P 



