Solving Mathematical Problems: Moose Population Growth and Equation Analysis

In this article, we will delve into two mathematical problems: determining the time for a moose population to grow to a specific number, and solving a complex equation with three unknowns. We will break down the problems and provide step-by-step solutions.

Moose Population Growth Analysis

The number of moose M on the island of Newfoundland is modeled by the equation M 75000 x 105-10t. Let us analyze how many years it will take for the moose population to grow to 100,000 animals.

Problem Statement

Given the equation:

M 75000 x 105-10t

Find t when M 100,000.

Solution

The moose population is increasing, so we can use the formula:

M Original population Increase in population

M 75000 x 105-10t

Set M 100,000 and solve for t:

100,000 75000 x 105-10t

Divide both sides by 75000:

100,000 / 75000 105-10t

1.3333 105-10t

Take the logarithm of both sides:

log(1.3333) (5 - 10t) log(10)

0.1249 5 - 10t

10t 5 - 0.1249

10t 4.8751

t 0.48751

Convert this to years, months, and days:

0.48751 x 12 5.846 months

0.846 x 30 25.38 days

t ≈ 0 years, 6 months, and 25 days

Equation Analysis with Three Unknowns

We will now solve a complex equation with three unknowns: n, a, and b.

Problem Statement

The following equations are given:

3 5 6 151872 5 5 6 253094 5 6 7 303585 5 5 3 251573

Determine the values of a, b, and n such that the equations are consistent.

Solution

First, we can start by solving for n:

3 5 6 151872

5 5 6 253094

5 6 7 303585

5 5 3 251573

Let us check for consistency:

2 - 1 0 0 101222

2 - 4 0 3 1521

3 - 2 0 1 1 50491

From these equations, we can solve for n, a, and b. We notice that the right-hand sides of the equations can be combined to form a solution.

Calculate n.

9 4 7 50611 * 9 49984 * 4 7 * 507

9 4 7 658984

Analysis

The best approach to solve this type of problem is to use observations and algebraic manipulation. We can test different values for n and see which one satisfies the equations. In this case, we use n 1 to simplify the equation and find the correct option.

By solving the equations, we find the correct value for n, a, and b.

Summary

Both the moose population growth problem and the equation analysis problem require careful manipulation and application of mathematical concepts. Understanding the problem, breaking it down into manageable steps, and using logical reasoning are key to finding the solutions.

Understanding these types of mathematical problems can improve your problem-solving skills and prepare you for more complex mathematical challenges.