Problem 10
Question
The U.S. population The Museum of Science in Boston displays a running total of the U.S. population. On May \(11,1993\) , the total was increasing at the rate of 1 person every 14 sec. The dis- played population figure for \(3 : 45\) P.M. that day was \(257,313,431\) . a. Assuming exponential growth at a constant rate, find the rate constant for the population's growth (people per 365 -day year). b. At this rate, what will the U.S. population be at \(3 : 45 \mathrm{PM}\) . Boston time on May \(11,2008 ?\)
Step-by-Step Solution
Verified Answer
The rate constant is about 0.0088 per year. The population will be approximately 293,578,133 in 2008.
1Step 1: Understanding Exponential Growth
Exponential growth can be modeled using the formula: \[ P(t) = P_0 \times e^{rt} \]where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate constant, and \( t \) is the time in years.
2Step 2: Determining Initial Growth Rate
We know the population increases by 1 person every 14 seconds. This implies a rate of growth of 1 person/14 seconds.Convert this to a yearly rate:\[ 365 \times 24 \times 60 \times 60 / 14 = 2,268,571.43 \text{ people per year} \] This is the population increase per year based on the given rate.
3Step 3: Calculating the Rate Constant
Given that the initial population \( P_0 = 257,313,431 \) and it increases by approximately 2,268,571 people in a year:Use the continuous growth formula to find the rate constant \( r \):\[ P(t) = P_0 \times e^{rt} P(t) = P_0 + ext{Annual Increase} = 257,313,431 + 2,268,571 259,581,999 = 257,313,431 \times e^{r} \text{Solving for } r: e^{r} = \frac{259,581,999}{257,313,431} r = \ln\left(\frac{259,581,999}{257,313,431}\right) \approx 0.0088\]
4Step 4: Find the Population in 2008
Using the rate constant \( r = 0.0088 \) and \( t = 2008 - 1993 = 15years\):\[ P(15) = 257,313,431 \times e^{0.0088 \times 15}\]Calculate:\[ P(15) \approx 257,313,431 \times e^{0.132}\]\[ P(15) \approx 257,313,431 \times 1.141\]\[ P(15) \approx 293,578,133 \]This would be the population after 15 years at the given exponential growth rate.
Key Concepts
Population GrowthGrowth Rate CalculationExponential Growth Formula
Population Growth
Population growth refers to the change in the number of individuals in a population over time. This concept is crucial for understanding how societies develop and plan for future needs like resources, infrastructure, and services. In our exercise, we are focusing on the growth of the U.S. population over time.
In real-world scenarios, population growth can occur due to various factors such as birth rates, death rates, immigration, and emigration. However, for the sake of our problem, we assume a simplified model where the growth is primarily determined by a constant birth rate. This allows us to apply mathematical models like exponential growth to predict future population numbers.
Considering exponential growth, it implies that the population grows by a constant rate each instant, leading to larger changes over time. This continuous compound growth highlights the significance of understanding population dynamics for effective planning and policy-making.
In real-world scenarios, population growth can occur due to various factors such as birth rates, death rates, immigration, and emigration. However, for the sake of our problem, we assume a simplified model where the growth is primarily determined by a constant birth rate. This allows us to apply mathematical models like exponential growth to predict future population numbers.
Considering exponential growth, it implies that the population grows by a constant rate each instant, leading to larger changes over time. This continuous compound growth highlights the significance of understanding population dynamics for effective planning and policy-making.
Growth Rate Calculation
Calculating the growth rate is essential for predicting future population size. The growth rate represents how fast the population is increasing over a specific time period. In our exercise, we initially know that the population increases by 1 person every 14 seconds.
To convert this to a yearly rate, we logically consider the number of seconds in a year. There are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. So, we multiply these together to get the total seconds in a year:
This calculated annual increase is critical for determining the exponential growth rate constant that shows how our initial population grows over a year.
To convert this to a yearly rate, we logically consider the number of seconds in a year. There are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. So, we multiply these together to get the total seconds in a year:
- 365 days/year x 24 hours/day x 60 minutes/hour x 60 seconds/minute
This calculated annual increase is critical for determining the exponential growth rate constant that shows how our initial population grows over a year.
Exponential Growth Formula
The exponential growth formula is a mathematical tool used to model situations where growth compounds continuously over time. It is given by:
In our scenario, we calculate the rate constant \(r\) by using the continuous growth formula. We know the population at the initial time \(P_0 = 257,313,431\) and the approximate increase per year. By using the exponential growth formula and solving for \(r\), we calculate that the growth rate constant is approximately \(0.0088\).
Once the rate constant is known, we can predict future population sizes. For instance, to find the population in 2008, which is 15 years after 1993, we use the derived rate constant and exponential growth formula to calculate the population at that future time, resulting in approximately 293,578,133 people. This demonstrates how powerful the exponential growth formula is for making long-term projections in population studies.
- \( P(t) = P_0 \times e^{rt} \)
In our scenario, we calculate the rate constant \(r\) by using the continuous growth formula. We know the population at the initial time \(P_0 = 257,313,431\) and the approximate increase per year. By using the exponential growth formula and solving for \(r\), we calculate that the growth rate constant is approximately \(0.0088\).
Once the rate constant is known, we can predict future population sizes. For instance, to find the population in 2008, which is 15 years after 1993, we use the derived rate constant and exponential growth formula to calculate the population at that future time, resulting in approximately 293,578,133 people. This demonstrates how powerful the exponential growth formula is for making long-term projections in population studies.
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