Problem 32 - Algebra 1: Concepts and Skills

Question

An initial population of 1000 starfish doubles each year for 4 years. What is the starfish population after 4 years?

Step-by-Step Solution

Answer

The starfish population will be 16,000 after 4 years of exponential growth.

Step 1: Identify the parameters

From the problem, it can be seen that the initial population \( P_0 = 1000 \), the number of years \( n = 4 \), and the population doubles each year which means the growth factor is 2.

Step 2: Apply the exponential growth formula

We use the formula: \( P = P_0 * 2^n \) . Substituting the values identified in Step 1 we have: \( P = 1000 * 2^4 \).

Step 3: Calculation

Next, calculate the result of the equation from Step 2: \( P = 1000 * 2^4 = 16000 \). So, the starfish population after 4 years would be 16,000.

Key Concepts

Exponential Growth in Algebra

When we talk about exponential growth in algebra, we're looking at situations where a quantity increases by a consistent rate over equal intervals of time. It’s like a snowball rolling down a hill; it starts small, but as it rolls, it picks up more snow, and its size grows exponentially. The general formula we use to calculate exponential growth is:
\[ P = P_0 \times r^n \]
where:

• \( P \) is the final amount after time \( n \).
• \( P_0 \) (pronounced 'P naught') is the initial amount.
• \( r \) is the growth rate, and
• \( n \) is the number of time periods that have passed.

In the exercise about the starfish, the population doubles each year. Doubling is a classic example of exponential growth, hence the use of the number 2 as the growth rate (\( r \)). To solve such problems, one must insert the specific values into the algebraic expression and calculate the result, as seen in the provided solution.

Population Growth Problems

Population growth problems are a practical application of exponential growth in the field of biology and environmental studies. They often involve organisms in an ecosystem, such as the population of starfish in our example. These problems highlight how populations can explode under ideal conditions where resources are abundant, and predators are minimal.

When solving these problems, understanding the contextual factors, like average lifespan, reproductive rate, and resource limitations, is crucial. However, in mathematical models, we simplify reality by assuming ideal growth conditions to demonstrate the potential for exponential increase. Thus, the formula \( P = P_0 \times r^n \) becomes an invaluable tool for predicting future population sizes. Calculating such growth allows ecologists to make informed decisions about conservation and management of species and habitats.

Algebraic Expressions and Equations

Algebraic expressions and equations are the building blocks of algebra, a branch of mathematics. An expression is a combination of numbers, variables (like \( x \) or \( n \)), and operations (such as addition or multiplication) that represents a value. An equation, on the other hand, is a statement that two expressions are equal, connected by an equals sign (\( = \)).

In our starfish population example, \( P_0 \times 2^n \) is an expression that represents the population after \( n \) years. When we write \( P = 1000 \times 2^4 \), it becomes an equation telling us the starfish population after 4 years is equal to 16,000. By understanding expressions and equations, students can manipulate and solve mathematical problems that model real-world scenarios, and learn to communicate their findings effectively.