At a certain remote lake, Maria has identified 2 particularly good locations to catch fish. At a location by a...
GMAT Two Part Analysis : (TPA) Questions
At a certain remote lake, Maria has identified \(\mathrm{2}\) particularly good locations to catch fish. At a location by a fallen tree, the expected yield from fishing is approximately \(\mathrm{1\,kg}\) of fish per hour, no matter how long she has been fishing there. At a nearby location near some lily pads, the expected yield is \(\mathrm{3\,kg}\) of fish in one hour, provided no one has been fishing there for at least a day. Each subsequent hour Maria spends fishing at the lily pads will have \(\frac{1}{2}\) the expected yield as that of the hour before it. Maria plans to fish from 4:00 to 10:00 tomorrow morning, each hour at one of these two locations, switching exactly one time (at the beginning of one of the hours).
On the assumption that no one else will be fishing at either location for several days before Maria's planned trip, select for Fallen tree and for Lily pads the numbers of hours Maria should spend at each of the two locations in order to maximize the expected yield from fishing during the time she plans to fish.
Phase 1: Owning the Dataset
Creating Our Visualization
We have a 6-hour fishing window (4:00 AM - 10:00 AM) and need to allocate time between two locations:
Timeline Visualization:
4:00 ----[Location 1]---- [Switch] ----[Location 2]---- 10:00
x hours (6-x) hours
Yield Characteristics:
- Fallen Tree: Constant \(1\) kg/hour
- Lily Pads: \(3\) kg (hour 1), \(1.5\) kg (hour 2), \(0.75\) kg (hour 3), \(0.375\) kg (hour 4), etc.
Let's calculate the lily pad yields for all 6 possible hours:
| Hour at Lily Pads | Yield (kg) |
| 1st hour | 3.000 |
| 2nd hour | 1.500 |
| 3rd hour | 0.750 |
| 4th hour | 0.375 |
| 5th hour | 0.188 |
| 6th hour | 0.094 |
Phase 2: Understanding the Question
We need to find how many hours Maria should spend at each location to maximize total fish yield.
Key Constraints:
- Total time = 6 hours
- Must switch exactly once
- Hours at each location must sum to 6
Critical Insight: The lily pads become less efficient than the fallen tree after hour 3 (\(0.75\) kg \(< 1\) kg). This suggests we should limit time at lily pads.
Phase 3: Finding the Answer
Let's systematically check total yields for different time allocations:
Starting at Lily Pads:
- 1 hour lily pads + 5 hours fallen tree: \(3 + (5 \times 1) = 8.00\) kg
- 2 hours lily pads + 4 hours fallen tree: \((3 + 1.5) + (4 \times 1) = 8.50\) kg ✓
- 3 hours lily pads + 3 hours fallen tree: \((3 + 1.5 + 0.75) + (3 \times 1) = 8.25\) kg
Starting at Fallen Tree:
- 4 hours fallen tree + 2 hours lily pads: \((4 \times 1) + (3 + 1.5) = 8.50\) kg ✓
- 5 hours fallen tree + 1 hour lily pads: \((5 \times 1) + 3 = 8.00\) kg
The maximum yield of \(8.50\) kg occurs when Maria spends:
- 4 hours at the Fallen Tree
- 2 hours at the Lily Pads
(Note: Both sequences achieve the same total, but the question asks for hours at each location, not the order.)
Phase 4: Solution
Final Answer:
- Fallen Tree: 4 hours
- Lily Pads: 2 hours
This allocation maximizes yield by capturing the high initial returns from lily pads (\(3 + 1.5 = 4.5\) kg) while avoiding the diminishing returns that fall below the fallen tree's consistent rate.