Problem 12 - Calculus

Question

The earth's population is growing at \(v=100\) million a year, starting from \(f=5.2\) billion in \(1990 .\) Graph \(f(t)\) and find \(f(2000)\)

Step-by-Step Solution

Answer

The population in 2000 is 6.2 billion.

Step 1: Understand the Problem

We are given a starting population of the Earth in 1990, which is 5.2 billion, and a constant rate of population increase of 100 million per year. We need to express this as a function of time and then find the population in the year 2000.

Step 2: Define the Population Function

The population function can be described as a linear equation because the population increases at a constant rate. Let \( t \) represent the number of years after 1990. The function is then \( f(t) = 5.2 + 0.1t \), where 5.2 is the initial population in billion and 0.1 is the annual growth rate in billion (since 100 million is 0.1 billion).

Step 3: Calculate Population in 2000

We need to find the population 10 years after 1990, so substitute \( t = 10 \) into our function: \( f(10) = 5.2 + 0.1 \times 10 \). This simplifies to \( f(10) = 5.2 + 1 = 6.2 \). Thus, the population in 2000 is 6.2 billion.

Step 4: Sketch the Graph

To graph \( f(t) = 5.2 + 0.1t \), plot the initial point (0, 5.2) and another point like (10, 6.2). Draw a line through these points, which represents the population growth over time. The line shows that the population grows linearly by 0.1 billion each year.

Key Concepts

Linear growth model

A linear growth model explains how a quantity changes at a consistent rate over time. It is a fundamental concept often used to represent population growth when the increase each year is the same. For example, if Earth’s population grows by 100 million people each year, this addition does not change, giving it a linear characteristic.
Linear growth is characterized by a straight line when graphed, indicating that the dependent variable, in this case, population, changes uniformly with respect to the independent variable, which is time.
In mathematical terms, a linear growth model can be expressed as a linear function of the form:\[ f(t) = mt + b \]Where:

• \( m \): Rate of change or growth (slope of the line)
• \( b \): Initial quantity (y-intercept)

In our example, the initial population is 5.2 billion and the linear growth rate is 0.1 billion (or 100 million).

Constant rate of change

A constant rate of change is a fixed amount that a quantity increases or decreases over equal increments of time. In our population growth scenario, this rate is 100 million people per year.
This means that every year, the population grows by exactly the same amount without any variation.
The term "constant" indicates that the rate does not change over time, which is crucial to maintaining linearity. The constant rate of change is what gives rise to a straight line in graphing since the slope remains the same across all points. It is the "m" in the equation \( f(t) = mt + b \), where it determines how steep the line is. For this population function, the constant rate is 0.1 billion.

Function of time

A function of time describes how a given quantity evolves as time progresses. Specifically, it relates time as the independent variable and another quantity, like population, as the dependent variable.
The population growth can be represented as a function of time with the equation \( f(t) = 5.2 + 0.1t \). Here:

• \( t \): Represents time in years since 1990
• \( f(t) \): Denotes the population at time \( t \) measured in billions

This function is linear, reflecting how population changes predictably over each unit of time without sudden spikes or reductions, aligned with the assumption of a constant rate of change.

Graphing linear functions

Graphing linear functions involves plotting a straight line on a coordinate plane where the line is defined by a specific linear equation. The purpose is to visually represent the relationship between the variables involved.
In the case of the population function \( f(t) = 5.2 + 0.1t \), the graph shows a direct and steady increase. It starts from an initial point, like (0, 5.2), reflecting the population in 1990, and moves upwards to other calculated points such as (10, 6.2) for the year 2000.
These points are plotted then connected with a line which illustrates the population's growth path. Each point on the line corresponds to the population for a specific year, and the nature of the line (straight) shows that the growth rate is uninterrupted and consistent.