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Two towns $\text{A}$ and $\text{B}$ are $100$ km apart. A school is to be built for $100$ students of town $\text{B}$ and $30$ students of Town $\text{A}$. Expenditure on transport is Rs. $1.20$ per km per student. If the total expenditure on transport by all $130$ students is to be as small as possible, then the school should be built at

- $33$ km from Town $\text{A}$
- $33$ km from Town $\text{B}$
- Town $\text{A}$
- Town $\text{B}$

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Let’s draw the diagram for a better understanding.

DIAGRAM 1

Let the school be at a distance $x$ km from town $A.$ Then the distance of school from $B = (100-x)$ km.

DIAGRAM 2

Now, the distance traveled by $30$ students from town $A$ to school $=30x.$

And, the distance traveled by $100$ students from town $B$ to school $=(100)(100-x) =(10000-100x).$

So, the total distance traveled by $130$ students $D=30x + 10000-100x = 10000-70x.$

$D= f(x) = 10000-70x $

If the total expenditure on transport by all $130$ students is to be small,then $x=0.$

$\therefore$ The school should be built at town $A.$

Correct Answer $: \text{C}$

DIAGRAM 1

Let the school be at a distance $x$ km from town $A.$ Then the distance of school from $B = (100-x)$ km.

DIAGRAM 2

Now, the distance traveled by $30$ students from town $A$ to school $=30x.$

And, the distance traveled by $100$ students from town $B$ to school $=(100)(100-x) =(10000-100x).$

So, the total distance traveled by $130$ students $D=30x + 10000-100x = 10000-70x.$

$D= f(x) = 10000-70x $

If the total expenditure on transport by all $130$ students is to be small,then $x=0.$

$\therefore$ The school should be built at town $A.$

Correct Answer $: \text{C}$