Question

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 be built is

• A. town B
• B. $45$ km from town A
• C. town A
• D. $45$ km from town B

Answer 1

Answer: (C)

town A

Clearly from figure, let f(x) be the distance travelled by the students
where x is the distance of the school from A
then
$f\left(x\right)=150x+50(60-x)$
$f\left(x\right)=100x+3000$
So,$f^{\prime }\left(x\right)=100>0$
Always increasing , so minimum at x = 0.
so school should be at A