Problem 126 - Elementary and Intermediate Algebra

Question

Suppose that a country with a population of 1,000 people is growing according to the formula \(P=1,000 e^{0.02 t}\) where \(t\) is in years. Furthermore, assume that the food supply \(F\), measured in adequate food per day per person, is growing linearly according to the formula \(F=30.625 t+2,000(t\) is time in years). Use a graphing calculator to determine in how many years the population will outstrip the food supply.

Step-by-Step Solution

Answer

The population will outstrip the food supply in approximately 39 years.

Step 1: Understand the Formulas

We are given two formulas: one for the population growth, \( P = 1,000 e^{0.02t} \), and one for the food supply growth, \( F = 30.625t + 2,000 \). We need to find when population \( P \) will exceed food supply \( F \).

Step 2: Set Up the Inequality

To find when the population exceeds the food supply, we need to solve the inequality: \( 1,000 e^{0.02t} > 30.625t + 2,000 \).

Step 3: Graph Both Functions

Using a graphing calculator, plot the functions \( P(t) = 1,000 e^{0.02t} \) and \( F(t) = 30.625t + 2,000 \). Aim to find the point where \( P \) first becomes greater than \( F \).

Step 4: Find the Intersection

On your graphing calculator, look for the point of intersection between the two graphs. This point will give the value of \( t \) where \( P = F \).

Step 5: Determine When Population Exceeds Food Supply

Identify the smallest integer \( t \) after this intersection point where \( P(t) > F(t) \). This is the year when the population will outstrip the food supply.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population over time. It is often modeled by exponential functions due to their natural tendency to multiply quickly. In the exercise provided, the population of a country is described by the formula \[ P = 1,000 e^{0.02t} \]where \( P \) is the population, and \( t \) is time in years. This is an example of exponential growth, characterized by constant relative growth rates.
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. In simpler terms, a small increase in population leads to an even larger subsequent increase. The key aspects of understanding exponential growth are:

• The base population or initial value (in this case, 1,000).
• The growth constant (here, 0.02), controlling how fast the population grows.

Exponential growth can lead to rapid increases, which sometimes creates challenges in things like resource allocation and infrastructure planning.

Linear Growth

Linear growth represents changes that happen at a constant rate over time. In contrast to exponential growth, linear growth describes a scenario where a quantity grows by the same amount per time period. This is often used to depict situations where resources like food or land are being developed or acquired steadily, rather than exponentially.
In our exercise, the food supply is modeled by the linear formula:\[ F = 30.625t + 2,000 \]where \( F \) is the food supply per day per person, and \( t \) is time in years.
The formula components include:

• The initial amount, represented by the intercept (2,000 in this case).
• The rate of change or slope (30.625 here), indicating how much the food supply increases per year.

Linear growth functions produce straight lines when graphed, showing steady change over time. It serves as a useful model for scenarios where conditions are stable, and the rate of growth doesn't change over the period considered.

Graphing Inequalities

When dealing with functions and real-world scenarios, graphing inequalities helps us visually represent and solve conditions where one quantity surpasses another. In this exercise, we explore when the population surpasses the food supply.
To solve such problems, graph both the population \( P(t) = 1,000 e^{0.02t} \) and the food supply \( F(t) = 30.625t + 2,000 \) functions using a graphing calculator. By visually comparing the graphs, you can find the point where the population starts to exceed the available food.
Key steps include:

• Identifying the intersection point, where \( P = F \).
• Determining the smallest integer \( t \) value where \( P(t) > F(t) \), after the intersection point.

Graphing inequalities is a vital skill, especially when tackling contrasting growth models like exponential and linear growth. This approach ensures we can foresee potential problems and plan accordingly to avoid resource shortages.