A tour operator charges ₹136 per passenger for 100 passengers with a discount of ₹4 for each 10 passengers in excess of 100. Find the number of passengers that will maximise the amount of money the tour operator receives.

220

The correct answer is Option (3) → 220

Let the number of passengers that are in excess of 100 be 10x, then amount received by the tour operator is

$A = (100+ 10x) (136 - 4x)$

$⇒A = 13600-400x + 1360x-40x^2$

$⇒A = 13600 + 960x - 40x^2$

Diff. w.r.t. x, we get

$\frac{dA}{dx}=0+960-80x$

and $\frac{d^2A}{dx^2}= -80$.

Now, $\frac{dA}{dx}=0⇒960-80x = 0⇒x= 12$.

Also, $\left[\frac{d^2A}{dx^2}\right]_{x=12}=-80<0$

⇒ A is maximum at $x = 12$.

Hence, the number of passengers for which the tour operator received maximum amount is $100 + 10 × 12$ i.e. 220.