IIT-JEE 1982Question 1
Two towns $A$ and $B$ are $60$ km apart. A school is to be built to serve $150$ students in town $A$ and $50$ students in town $B$. If the total distance to be travelled by all $200$ students is to be as small as possible, then the school should be built at
Differentiation and its Application
Solution
Concept. Put the school at distance $d$ from $A$ on the road to $B$; the total distance is linear in $d$. Formulas used. (1) A linear function on $[0,60]$ is least at an end pointStep 1 - Total distance. $D(d)=150d+50(60-d)=3000+100d$. Step 2 - Minimise. $D$ increases with $d$, so by (1) the least value is at $d=0$: in town $A$. Answer: (C)