Problem 84
Question
The development of AstroWorld ("The Amusement Park of the Future") on the outskirts of a city will increase the city's population at the rate of $$ 4500 \sqrt{t}+1000 $$ people/year, \(t\) yr from the start of construction. The population before construction is \(30,000 .\) Determine the projected population 9 yr after construction of the park has begun.
Step-by-Step Solution
Verified Answer
The projected population of the city 9 years after the construction of AstroWorld has begun is 79,500 people.
1Step 1: Integrate the given function
To find the population at any time t, we need to integrate the given function w.r.t t:
$$
\int (4500 \sqrt{t} + 1000) dt
$$
2Step 2: Apply the power rule of integration
We apply the power rule which states that \(\int{x^n} dx = \frac{x^{n+1}}{n+1}+ C\).
So, for our function, after integrating, we get:
$$
\left(\frac{4500t^{(1/2 + 1)}}{1/2 + 1}\right) + (1000t) + C
$$
3Step 3: Simplify the integrated function
After simplifying the function, we have:
$$
\left(\frac{4500t^{3/2}}{3/2}\right) + (1000t) + C
$$
4Step 4: Find the value of C with the initial population
We know the initial population was 30,000 when t = 0. So, to find C, we substitute the values of P(t) and t:
$$
30000 = \left(\frac{4500(0)^{3/2}}{3/2}\right) + (1000(0)) + C
$$
From the equation above, we find that C = 30,000. So, our population function is:
$$
P(t) = \left(\frac{4500t^{3/2}}{3/2}\right) + (1000t) + 30000
$$
5Step 5: Find the population after 9 years
Now, we can use the population function to find the projected population when t = 9 years:
$$
P(9) = \left(\frac{4500(9)^{3/2}}{3/2}\right) + (1000(9)) + 30000
$$
After calculating, we get:
$$
P(9) = 79500
$$
So, the projected population of the city 9 years after the construction of AstroWorld has begun is 79,500 people.
Key Concepts
Population Growth ModelingPower Rule of IntegrationDefinite Integrals
Population Growth Modeling
Population growth modeling is a crucial tool used to predict how a population will change over time in response to various factors. In this exercise, the growth of a city's population is influenced by the construction of an amusement park. The rate at which the population increases is given by the function \(4500 \sqrt{t} + 1000 \text{ people/year}\), where \(t\) represents time in years from the start of construction.
This model helps in understanding how an external factor, like the amusement park, can spur growth. By knowing the initial population and the rate of increase, we can project future population sizes. Through the integration process, we convert a rate of change into a function for population, allowing us to calculate population at any given time. Here, the initial population was 30,000 people. Using our projected model, we could determine the city's population will reach 79,500 in 9 years after construction began.
This model helps in understanding how an external factor, like the amusement park, can spur growth. By knowing the initial population and the rate of increase, we can project future population sizes. Through the integration process, we convert a rate of change into a function for population, allowing us to calculate population at any given time. Here, the initial population was 30,000 people. Using our projected model, we could determine the city's population will reach 79,500 in 9 years after construction began.
Power Rule of Integration
The power rule of integration is a fundamental rule of calculus that simplifies the integration of functions involving powers of a variable. This rule states that if you have a function of the form \(x^n\), its integral is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
In our problem, the power rule is applied to \(4500 \sqrt{t}\). Recognizing \(\sqrt{t}\) as \(t^{1/2}\), we use the power rule to find its integral:
The power rule simplifies complex integration tasks, transforming rates of change into holistic growth models.
In our problem, the power rule is applied to \(4500 \sqrt{t}\). Recognizing \(\sqrt{t}\) as \(t^{1/2}\), we use the power rule to find its integral:
- Integrate \(4500 t^{1/2}\) using: \(\frac{4500 t^{3/2}}{3/2}\).
The power rule simplifies complex integration tasks, transforming rates of change into holistic growth models.
Definite Integrals
Definite integrals are used to calculate the accumulated quantity, such as an increase in population over a specified time period. In this context, we've used a definite integral approach to understand how the city's population changes over 9 years.
While indefinite integrals are necessary to form the general solution with a constant \(C\), definite integrals allow us to determine actual numbers by subtracting initial values from final values over a given interval. However, in our example, the process closely mimics definite integration as we're asked to calculate growth from \(t=0\) to \(t=9\) years directly by using the derived population function \(P(t)\).
By applying definite values into the final population function, we successfully find the population at 9 years, showing a tangible application of definite integrals through calculus integration.
While indefinite integrals are necessary to form the general solution with a constant \(C\), definite integrals allow us to determine actual numbers by subtracting initial values from final values over a given interval. However, in our example, the process closely mimics definite integration as we're asked to calculate growth from \(t=0\) to \(t=9\) years directly by using the derived population function \(P(t)\).
By applying definite values into the final population function, we successfully find the population at 9 years, showing a tangible application of definite integrals through calculus integration.
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