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Routes and Networks
The table below shows the strictly one-way connections between various nodes in a delivery network, starting at the Source (S) and ending at the Destination (**T**). For each route segment, the travel time (in minutes) and the toll cost (in USD) are provided. $$\begin{array}{|c|c|c|} \hline \textbf{Route Segment} & \textbf{Time (minutes)} & \textbf{Toll Cost (USD)} \\ \hline \text{S} \rightarrow \text{P} & 15 & 10 \\ \hline \text{S} \rightarrow \text{Q} & 10 & 40 \\ \hline \text{P} \rightarrow \text{Q} & 5 & 10 \\ \hline \text{P} \rightarrow \text{R} & 25 & 20 \\ \hline \text{Q} \rightarrow \text{R} & 10 & 5 \\ \hline \text{Q} \rightarrow \text{T} & 30 & 30 \\ \hline \text{R} \rightarrow \text{T} & 15 & 15 \\ \hline \end{array}$$ A transporter earns a flat base revenue of **100 USD** for successfully completing the delivery from S to T. The net profit for the delivery is calculated using the following formula: $$\text{Net Profit} = \text{Base Revenue} - \text{Total Toll Cost} - \text{Time Penalty}$$ The **Time Penalty** is applied strictly if the total travel time exceeds **40 minutes**. For every additional minute beyond 40 minutes, a penalty of **3 USD per minute** is deducted. Assuming the transporter chooses the route that maximizes their Net Profit, what is the maximum possible Net Profit? A) 40 USD B) 45 USD C) 50 USD D) 35 USD
**Correct Answer: B) 45 USD** ### **Step-by-Step Solution** **Step 1: Identify all possible routes from Source (S) to Destination (T)** By tracing the nodes forward from S, we can map out exactly 5 distinct paths to T: 1. S $\rightarrow$ P $\rightarrow$ R $\rightarrow$ T 2. S $\rightarrow$ P $\rightarrow$ Q $\rightarrow$ R $\rightarrow$ T 3. S $\rightarrow$ P $\rightarrow$ Q $\rightarrow$ T 4. S $\rightarrow$ Q $\rightarrow$ R $\rightarrow$ T 5. S $\rightarrow$ Q $\rightarrow$ T **Step 2: Calculate Time, Cost, and Penalty for each route** The time penalty formula mathematically is: $3 \times \max(0, \text{Total Time} - 40)$. * **Path 1:** S $\rightarrow$ P $\rightarrow$ R $\rightarrow$ T * **Time:** $15 + 25 + 15 =$ **55 mins** * **Toll:** $10 + 20 + 15 =$ **45 USD** * **Penalty:** $(55 - 40) \times 3 =$ **45 USD** * **Profit:** $100 - 45 - 45 =$ **10 USD** * **Path 2:** S $\rightarrow$ P $\rightarrow$ Q $\rightarrow$ R $\rightarrow$ T * **Time:** $15 + 5 + 10 + 15 =$ **45 mins** * **Toll:** $10 + 10 + 5 + 15 =$ **40 USD** * **Penalty:** $(45 - 40) \times 3 =$ **15 USD** * **Profit:** $100 - 40 - 15 =$ **45 USD** * **Path 3:** S $\rightarrow$ P $\rightarrow$ Q $\rightarrow$ T * **Time:** $15 + 5 + 30 =$ **50 mins** * **Toll:** $10 + 10 + 30 =$ **50 USD** * **Penalty:** $(50 - 40) \times 3 =$ **30 USD** * **Profit:** $100 - 50 - 30 =$ **20 USD** * **Path 4:** S $\rightarrow$ Q $\rightarrow$ R $\rightarrow$ T * **Time:** $10 + 10 + 15 =$ **35 mins** * **Toll:** $40 + 5 + 15 =$ **60 USD** * **Penalty:** 35 mins is under the 40 min threshold $=$ **0 USD** * **Profit:** $100 - 60 - 0 =$ **40 USD** * **Path 5:** S $\rightarrow$ Q $\rightarrow$ T * **Time:** $10 + 30 =$ **40 mins** * **Toll:** $40 + 30 =$ **70 USD** * **Penalty:** 40 mins exactly hits the threshold $=$ **0 USD** * **Profit:** $100 - 70 - 0 =$ **30 USD** **Step 3: Conclusion** Comparing the net profits (10, 45, 20, 40, 30), the transporter maximizes their earnings by choosing **Path 2**, resulting in a maximum profit of **45 USD**. *(Note for CAT prep: Notice how the optimal path wasn't the fastest one, nor was it the one with the fewest nodes. CAT paper setters love rewarding students who systematically test "longer" topological routes that balance competing constraints!)*
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