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: Understanding the Concept:
Revenue here is a quadratic function of the number of extra travellers, and a quadratic with a negative leading coefficient always has a single highest point. We can find that highest point directly by completing the square instead of using the vertex formula.

Step 2: Key Formula or Approach:
Let k be the number of travellers past 200. Group size is \(200+k\), price per person is \(20000-50k\), so revenue is $R(k) = (200+k)(20000-50k)$, which expands to $R(k) = -50k^2 + 10000k + 4{,}000{,}000$.

Step 3: Detailed Explanation:
Factor out $-50$ from the k terms:
$R(k) = -50(k^2 - 200k) + 4{,}000{,}000$
Complete the square inside the bracket: $k^2 - 200k = (k-100)^2 - 10000$. Substituting back:
$R(k) = -50[(k-100)^2 - 10000] + 4{,}000{,}000$
$R(k) = -50(k-100)^2 + 500{,}000 + 4{,}000{,}000$
$R(k) = 4{,}500{,}000 - 50(k-100)^2$
Since $(k-100)^2$ is a square, it is never negative, so the term $-50(k-100)^2$ is never positive. This means $R(k)$ can never exceed $4{,}500{,}000$, and it hits that value exactly when $k = 100$, meaning a group of 300 people paying Rs. 15,000 each.

Step 4: Final Answer:
The maximum possible revenue Raj Travels can earn from this package is Rs. 45,00,000.
\[ \boxed{Rs. 45,00,000} \]