Solving a Moose Population Growth Problem and Analyzing Computational Efficiency

Understanding the dynamics of a moose population on an island can provide valuable insights into conservation efforts and ecosystem management. Let's explore a specific problem involving the moose population on the island of Newfoundland.

Problem Statement and Context

The moose population is modeled by the function M 75,000 cdot 10^{5-frac{t}{5}}, where t is time in years. The goal is to determine the number of years it will take for the moose population to grow to 100,000 animals.

Solution and Calculation

To find the time t required for the moose population to reach 100,000 animals, we can use the given formula:

100,000 75,000 cdot 10^{5-frac{t}{5}}

First, simplify the equation:

[ frac{100,000}{75,000} 10^{5-frac{t}{5}} ]

[ frac{4}{3} 10^{5-frac{t}{5}} ]

Next, take the logarithm (base 10) of both sides:

[ log_{10}left(frac{4}{3}right) 5-frac{t}{5} ]

[ frac{t}{5} 5 - log_{10}left(frac{4}{3}right) ]

[ t 25 - 5 log_{10}left(frac{4}{3}right) ]

Using a calculator or computational tool, we find:

[ t approx 25 - 5 times 0.1249 25 - 0.6245 24.3755 ]

Therefore, it will take approximately 24.3755 years for the moose population to grow to 100,000 animals.

Computational Efficiency and Solutions

While the manual calculation provides a clear understanding of the problem, it is also important to consider computational efficiency. The linked page discusses multiple solutions to the problem, including ones that run in O(n) time using O(1) auxiliary space. These solutions are particularly valuable in scenarios where computational resources are limited.

Here's a brief overview of the solutions mentioned:

One solution involves iterating over the array of heights and maintaining the maximum height to the left and right of each bar. Another solution uses a stack to efficiently calculate the trapped rainwater, which runs in O(n) time and O(n) space.

Additional Insight: Rainwater Trapping Problem

Let's solve a variation of the rainwater trapping problem. The objective is to compute the amount of water that can be trapped.

Python Script for Rainwater Trapping

def rainWater(arr):
    n  len(arr)
    L  [0] * n  # max height to the left
    R  [0] * n  # max height to the right
    T  [0] * n  # amount of water trapped at this bar
    m  0
    for i in range(n):
        m  max(m, arr[i])
        L[i]  m
    m  0
    for i in range(n-1, -1, -1):
        m  max(m, arr[i])
        R[i]  m
    for i in range(n):
        T[i]  min(L[i], R[i]) - arr[i]
    return sum(T)
input Arrays for testing:
arr  [[2, 0, 2], [3, 0, 2, 0, 4], [0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1]]
for E in arr:
    print(E)
    print(rainWater(E))

This script defines a function to calculate the trapped rainwater for an array of heights. It computes the maximum height to the left and right of each bar and calculates the trapped water accordingly.

Conclusion

Understanding the dynamics of populations, such as the moose population on Newfoundland, helps in making informed decisions about conservation efforts. Similarly, understanding efficient computational methods, like those used in solving the rainwater trapping problem, is essential for managing and optimizing resources and algorithms.

References:

Moore, M. (n.d.). *Moose population growth problem and solution*. GeeksforGeeks. (n.d.). *Rainwater Trapping Problem*.