Problem 48 - Applied Calculus

Question

In 2010 , there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US. \(^{70}\) The number of vehicles grew \(15.5 \%\) over the previous decade, while the population has been growing at \(9.7 \%\) per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?

Step-by-Step Solution

Answer

There will be one vehicle per person by approximately 2040.

Step 1: Understand the Growth Model

We start by understanding that both the number of vehicles and the population grow exponentially. This means we can use the exponential growth formula for both, i.e., \( N(t) = N_0 \times (1 + r)^t \), where \( N(t) \) is the amount after \( t \) decades, \( N_0 \) is the initial amount in 2010, and \( r \) is the rate of growth per decade.

Step 2: Vehicle Growth Equation

The number of vehicles in 2010 was 246 million, with a growth rate of 15.5%. Therefore, the growth equation for vehicles is \( V(t) = 246 \times (1 + 0.155)^t \).

Step 3: Population Growth Equation

The population in 2010 was 308.7 million, with a growth rate of 9.7%. Therefore, the growth equation for the population is \( P(t) = 308.7 \times (1 + 0.097)^t \).

Step 4: Set Up the Equation for Competing Ratios

We want the number of vehicles to match the number of people. Therefore, we set up the equation \( 246 \times (1 + 0.155)^t = 308.7 \times (1 + 0.097)^t \).

Step 5: Simplify the Equation

Divide both sides of the equation by 246, and rearrange terms: \( (1 + 0.155)^t = \frac{308.7}{246} \times (1 + 0.097)^t \).

Step 6: Solve for \(t\)

Simplify and solve for \(t\):\[\left(\frac{1.155}{1.097}\right)^t = \frac{308.7}{246}\]Taking the natural log of both sides gives:\[t \cdot \ln\left(\frac{1.155}{1.097}\right) = \ln\left(\frac{308.7}{246}\right)\]Solve for \(t\):\[t = \frac{\ln\left(\frac{308.7}{246}\right)}{\ln\left(\frac{1.155}{1.097}\right)} \approx 2.9\]

Step 7: Interpret the Result

The solution \(t \approx 2.9\) implies that it will take almost three decades from 2010 for the number of vehicles to match the population. Thus, one vehicle per person will be achieved around the year 2040.

Key Concepts

Vehicle Population Ratio

The concept of Vehicle Population Ratio helps us understand the relationship between the number of vehicles and the number of people in a given region. It gives a clear picture of how vehicle ownership and population size compare.
In our exercise, we're trying to find when the number of vehicles per person will reach a 1:1 ratio in the US. This involves assessing how both the total number of vehicles and the population grow over time.
This metric is crucial as it impacts infrastructure, environmental policies, and city planning. By knowing how fast the number of vehicles grows relative to the population, governments can better plan for future transportation needs.
For accurate predictions, one must consider factors such as growth rates and initial numbers to comprehensively determine when the vehicle-to-people ratio reaches 1:1. In this scenario, both grow exponentially, which requires careful calculation to forecast the crossover point.

Growth Rate Calculation

Growth Rate Calculation is essential in determining how quickly a particular metric is increasing over time. In the context of vehicle and population growth, understanding this rate helps predict future needs and changes.
To calculate growth rates accurately, you need an initial value, the growth rate percentage, and the duration over which growth occurs. Exponential growth rates are often expressed in a percentage per decade, as seen in our case exercise.
For the vehicle growth, the equation used was:

• Initial vehicles: 246 million
• Growth rate: 15.5% per decade

This means the number of vehicles grows by 15.5% every ten years. Similarly, the population growth rate considered was 9.7% per decade. By applying these rates to the exponential growth formula, we can project future numbers.
It is crucial to solve for the time (t) at which two growth equations yield the same result—in this case, the point when each person has one vehicle.

Exponential Growth Equation

Exponential Growth Equation is the mathematical foundation for calculating how populations or numbers increase over time at a steady growth rate. This type of growth implies that the larger the number becomes, the faster it grows.
The equation used is:

• \[ N(t) = N_0 imes (1 + r)^t \]
• where \( N(t) \) is the future value, \( N_0 \) is the initial amount, and \( r \) is the growth rate.

In our exercise, the future population and vehicle numbers are calculated using this formula. By plugging in the respective initial values and growth rates, we can predict their future sizes.
The challenging part is to find when the number of vehicles will equal the number of people. This involves solving an equation where both elements are growing exponentially but at different rates. The magic happens when both sides of the equation are set equal to each other, allowing us to determine the time frame where they match.