Problem 79 - Elementary and Intermediate Algebra

Question

The 20th Century. The exponential function \(A(t)=123 e^{0.0117 t}\) approximates the population of the United States (in millions), where \(t\) is the number of years after 1930 . Use the function to estimate the U.S. population for these important dates: 1937 The Golden Gate Bridge is completed 1941 The United States enters World War II 1955 Rosa Parks refuses to give up her seat on a Montgomery, Alabama, bus 1969 Astronaut Neil Armstrong walks on the moon 1974 President Nixon resigns 1986 The Challenger space shuttle explodes 1997 The Simpsons becomes the longest running cartoon television series in history

Step-by-Step Solution

Answer

Use the given function and substitute each year to estimate the U.S. population for the specified dates.

Step 1: Understanding the Problem

We need to estimate the U.S. population for specific years using the function \(A(t)=123 e^{0.0117 t}\). The variable \(t\) is the number of years after 1930. For each important date, we calculate \(t\) and use it in the exponential formula to find \(A(t)\).

Step 2: Calculating Population for 1937

To find the population in 1937, calculate \(t = 1937 - 1930 = 7\). Substitute \(t\) into the equation: \[ A(7) = 123 e^{0.0117 imes 7} \] Calculate this to find the population for 1937.

Step 3: Calculating Population for 1941

Calculate for \(t = 1941 - 1930 = 11\). Substitute \(t\) into the equation: \[ A(11) = 123 e^{0.0117 imes 11} \] Compute this value to find the population for 1941.

Step 4: Calculating Population for 1955

Determine \(t = 1955 - 1930 = 25\). Use this in the equation: \[ A(25) = 123 e^{0.0117 imes 25} \] Evaluate to estimate the population for 1955.

Step 5: Calculating Population for 1969

Here, \(t = 1969 - 1930 = 39\). Replace \(t\) in the equation: \[ A(39) = 123 e^{0.0117 imes 39} \] Complete the calculation for the population estimate of 1969.

Step 6: Calculating Population for 1974

Use \(t = 1974 - 1930 = 44\). Insert \(t\) into the equation: \[ A(44) = 123 e^{0.0117 imes 44} \] Determine the population for 1974.

Step 7: Calculating Population for 1986

Find \(t = 1986 - 1930 = 56\). Plug this into the formula: \[ A(56) = 123 e^{0.0117 imes 56} \] Compute this to estimate the population for 1986.

Step 8: Calculating Population for 1997

Compute \(t = 1997 - 1930 = 67\). Substitute \(t\) back into the equation: \[ A(67) = 123 e^{0.0117 imes 67} \] Finish the calculation to find the population for 1997.

Step 9: Summary of Results

After performing the calculations, gather the population estimates for each year. This can be done with a calculator to yield a numerical value for each \(A(t)\) calculated.

Key Concepts

Population Estimation

When we talk about population estimation, it often involves using models to predict the number of people in a specific region at a given time. In this exercise example, we estimate the United States population during various historically significant years of the 20th century.
Population estimation is crucial because it helps:

• Governments plan for future needs like infrastructure and resources.
• Organizations predict market trends and allocate resources efficiently.
• Research institutions study demographic shifts and their impacts on society.

To perform these estimations, we often rely on mathematical models like exponential functions, which can describe how populations grow over time. This allows us to make informed predictions even with just a few data points.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena with mathematical formulas. In our case, we deal with modeling the population growth of a country via an exponential function. The function provided in this exercise, \( A(t) = 123 e^{0.0117 t} \), is a mathematical model.
With mathematical models:

• We simplify complex systems into understandable equations.
• Gain insights about future trends based on historical data.
• Can create predictive tools that support decision-making and planning.

This particular model assumes an exponential growth, implying that the population grows at a rate proportional to its size. In other words, a larger population grows more quickly than a smaller one given the same conditions.

Algebraic Equations

Algebraic equations are foundational in expressing relationships between variables using mathematical symbols. In this exercise, we use the equation \( A(t) = 123 e^{0.0117 t} \) to determine the population at specific times.
Exponential functions are a special type of algebraic equation that graph as a rising curve. They have the form \( y = a e^{bx} \), where \( a \) represents the base population and \( b \) is the growth rate in this context.
To solve these algebraic equations, we:

• Identify the variable \( t \) which marks the number of years after 1930.
• Substitute specific values of \( t \) into the equation to find \( A(t) \), the estimated population.
• Calculate \( A(t) \) using a calculator or computational tool for accurate results.

This gives us the capability to predict population sizes for various future dates, providing essential information derived from somewhat simple computations.