Question

Raj Travels has the following revenue model for a group package. The owner charges Rs. 20,000 per person for a group size up to 200. For every additional traveller beyond 200, he starts offering a discount of Rs. 50 to ALL members of the group.

The maximum possible income for Raj Travels from the package is:

• A. Rs. 4000000
• B. Rs. 4200000
• C. Rs. 4500000
• D. Rs. 5000000

Hint:

Write revenue as a quadratic function of the number of extra travellers beyond 200, then find where that quadratic peaks.

Answer 1

The Correct Option is C

Step 1: Set up a variable.
Let k be the number of travellers beyond 200 (so k can be 0, 1, 2, and so on). The group size is then \( n = 200 + k \). Because of the discount rule, the price charged per person is \( 20000 - 50k \), and this reduced price applies to every single member of the group, not just the extra ones.

Step 2: Write revenue as a function of k.
Total revenue is group size times price per person:
\[ R(k) = (200+k)(20000-50k) \]
Step 3: Expand the expression.
\[ R(k) = 200(20000) + 200(-50k) + k(20000) + k(-50k) \] \[ R(k) = 4{,}000{,}000 - 10{,}000k + 20{,}000k - 50k^2 \] \[ R(k) = 4{,}000{,}000 + 10{,}000k - 50k^2 \]
Step 4: Maximize this quadratic.
The coefficient of \(k^2\) is negative, so this is a downward opening parabola in k, and it peaks at \( k = -\dfrac{b}{2a} = -\dfrac{10000}{2(-50)} = 100 \).

Step 5: Compute the peak revenue.
\[ R(100) = 4{,}000{,}000 + 10{,}000(100) - 50(100)^2 = 4{,}000{,}000 + 1{,}000{,}000 - 500{,}000 = 4{,}500{,}000 \]
Step 6: Check this against the boundary cases.
At k = 0 (group of 200, no discount at all), revenue is \( R(0) = 4{,}000{,}000 \), which matches option A but is lower than the peak. At k = 200 (group of 400, price down to Rs. 10,000), revenue is \( R(200) = 4{,}000{,}000 + 2{,}000{,}000 - 2{,}000{,}000 = 4{,}000{,}000 \), the same as the start and still lower than the peak. This confirms k = 100 (a group of 300 people paying Rs. 15,000 each) gives the true maximum, ruling out the larger options D and E as unreachable.

Final Answer:
The maximum possible income is
\[ \boxed{\text{Rs. } 45,00,000} \]