An event management company charges ₹4800 per guest, for a bulk booking for 100 guests. In addition, it offers a discount of ₹200 for each group of 10 guests over and above 100 guests booking. What is the number of guests that will maximize the amount of money the company receives on a booking? What is the maximum profit on such booking?

Number of guests = 170, Maximum Profit = ₹5,78,000

The correct answer is Option (2) → Number of guests = 170, Maximum Profit = ₹5,78,000

Let the number of guests for the booking be x.

Clearly, $x > 100$.

According to given, profit $(P) = \left[4800-\frac{200}{10}(x - 100)\right] x$

$P=6800x-20x^2$   ...(i)

Differentiating (i) w.r.t. x, we get

$\frac{dP}{dx}=6800-40x$ and $\frac{d^2P}{dx^2}= -40$.

Now, $\frac{dP}{dx}=0⇒ 6800-40x = 0⇒x=170$

$\frac{d^2P}{dx^2}=-40 <0$  $⇒ P$ is maximum at $x = 170$.

Hence, a booking of 170 guests will maximize the profit and maximum profit $= 6800 × 170 - 20 × (170)^2 = ₹5,78,000$.