Problem 28 - Algebra 1

Question

In 1970 the population \(A\) of Arizona was \(1,775,000\). since then, the average annual percent of increase has been about \(3.41 \%\) Which model best fits the situation where \(t\) is the number of years since \(1970 ?\) (A) \(A=3.41 t+1.775,000\) (B) \(A=60,527.5 t+1,775,000\) (C) \(A=1,835,527.5 t^{2}\) (D) \(A=1,775,000(1.0341)^{t}\)

Step-by-Step Solution

Answer

The model that best fits the situation is (D) \(A=1,775,000*(1.0341)^{t}\)

Step 1: Understand the exponential growth model

A typical exponential growth model has the form \(P=a*(1+r)^{t}\), where: \n- \(P\) is the final amount or quantity \n- \(a\) is the initial amount or quantity (the amount or quantity at the beginning of the time frame being considered) \n- \(r\) is the growth rate per period (expressed as a decimal) \n- \(t\) is the time it takes, expressed in the same units of time as r.

Step 2: Identify the information given

In this problem, we are given that the initial population (a) is 1,775,000. The growth rate (r) is 3.41%, or 0.0341 when expressed as a decimal, and t is the time since 1970 in years. The function \(A(t)\) represents the population at a given time \(t\).

Step 3: Identify the correct model

Based on steps 1 and 2, we can match the form of the function described in Step 1 with the answer choices given. Choice (D), \(A=1,775,000*(1.0341)^{t}\), fits the form of the exponential growth function perfectly. Thus, (D) is the correct choice.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model scenarios where growth or decay happens at a rate proportional to the current value. They are essential for describing processes that increase or decrease rapidly.

For example, the formula for an exponential function that models growth is often written as \(A(t) = a \times (1 + r)^t\), where:

• \(A(t)\) represents the amount at time \(t\),
• \(a\) is the initial amount,
• \(r\) is the growth rate, and
• \(t\) is time.

The growth rate, \(r\), is typically expressed as a decimal, so a 3.41% growth rate would be entered into the formula as 0.0341.

Population Growth

Population growth is the increase in the number of individuals in a population and can often be modeled by exponential functions when the growth rate is constant. Real-world population growth can be influenced by many factors including birth rates, death rates, and migration, but simple models often assume a constant rate of growth.

To predict future populations, it's crucial to have a starting value and a consistent growth rate. The exponential growth formula used for populations is similar to that used for other exponentially growing quantities and can help demographers, scientists, and policy-makers make essential decisions based on predicted population sizes.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures and language to study and solve real-world problems. Models help to predict and explain behaviors and patterns by using assumptions and known data. In the case of exponential population growth, the model helps to predict future population sizes.

Creating a model requires both understanding the situation being modeled and translating that understanding into mathematical form. With the given example on Arizona's population growth, using historical data and the average annual percent increase, a mathematical model can project future population size. Models are not perfect replicas of reality, but they provide a useful approximation that can guide decision-making and planning.