The cost of running a bus from A to B is Rs. | vedclass

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Question

(C) Let the cost function be $C(v) = av + \frac{b}{v}$.Given that at $v = 30 	ext{ km/h}$,$C = 75$,so $30a + \frac{b}{30} = 75 \implies 900a + b = 22

Vedclass · Accepted Answer

$60$

Vedclass · Answer

(C) Let the cost function be $C(v) = av + \frac{b}{v}$.Given that at $v = 30 	ext{ km/h}$,$C = 75$,so $30a + \frac{b}{30} = 75 \implies 900a + b = 2250 \quad (i)$.Given that at $v = 40 	ext{ km/h}$,$C = 65$,so $40a + \frac{b}{40} = 65 \implies 1600a + b = 2600 \quad (ii)$.Subtracting $(i)$ from $(ii)$,we get $700a = 350$,which implies $a = 0.5$.Substituting $a = 0.5$ into $(i)$,we get $900(0.5) + b = 2250 \implies 450 + b = 2250 \implies b = 1800$.To find the most economical speed,we minimize $C(v)$ by setting $\frac{dC}{dv} = 0$.$\frac{dC}{dv} = a - \frac{b}{v^2} = 0 \implies v^2 = \frac{b}{a}$.$v^2 = \frac{1800}{0.5} = 3600 \implies v = 60 	ext{ km/h}$.

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