Answer:
[tex]\boxed{7588; \text{d. the population increase}}[/tex]
Step-by-step explanation:
[tex]b(t) = 2100e^{0.023t}\\d(t) = 1470e^{0.017t}[/tex]
1. Area between the curves
In the graph below, the birth curve is the upper exponential curve, and the death curve is the lower one
[tex]A = \displaystyle \int_{0}^{10}[b(t) - d(t)]dt\\\\A = \displaystyle \int_{0}^{10}[2100e^{0.023t} - 1470e^{0.017t}]dt\\\\A = \left [ \dfrac{2100}{0.023}e^{0.023t} - \dfrac{1470}{0.017 }e^{0.017t}\right ]_{0}^{10}\\\\[/tex]
[tex]A = \left [ 91304e^{0.023t} - 86471e^{0.017t}\right ]_{0}^{10}\\\\A = \left [91304e^{0.23} - 86471e^{0.17} \right ] - [91304- 86471] \\\\A = 91304 \times 1.259 - 86471\times 1.185 - 4834 = 114916 - 102494 - 4834 = \mathbf{7588}\\\\\text{The area between the two curves is $\boxed{\mathbf{7588}}$}[/tex]
2. Meaning of the area
[tex]\text{b(t) - f(t) = birth rate -death rate = rate of population increase} = \frac{\text{d}P}{\text{d}t}\\\\\int \frac{\text{d}P}{\text{d}t} \text{d}t=P\\\\\text{The integral represents $\boxed{\textbf{the population increase}}$ over ten years}[/tex]
The difference between the birth rate and the death rate of a population is the natural growth rate of the population
Part I: The area between the curves for 0 ≤ t ≤ 10 is approximately 7,587
Part II: The correct option is option d.
d. This area represents the increase in population over a 10-year period
The reasons the value and selected option are correct includes;
The given parameters are;
Birth rate function b(t) = [tex]2,100 \cdot e^{0.023 \cdot t}[/tex]
Dirth rate function, d(t) = [tex]1,470\cdot e^{0.017 \cdot t}[/tex]
Part I: Required;
To find the area between the curves for 0 ≤ t ≤ 10 (answer given to the nearest integer)
Solution:
The area between the curves, A, is given as follows;
[tex]A = \displaystyle \int\limits^{10}_0 {2,100 \cdot e^{0.023 \cdot t}-1,470\cdot e^{0.017 \cdot t}} \, dt = \left[\frac{2,100}{0.023} \cdot e^{0.023 \cdot t} - \frac{1,470}{0.017} \cdot e^{0.017 \cdot t}\right]^{10}_0[/tex]
[tex]\displaystyle A = \left[\frac{2,100}{0.023} \cdot e^{0.023 \cdot t} - \frac{1,470}{0.017} \cdot e^{0.017 \cdot t}\right]^{10}_0 \approx 7,587.9[/tex]
Given that the calculation is with regards to population which are whole numbers, the value is rounded down as A ≈ 7,587
Part II: Required:
To state what the area in part (a) represents
Solution:
The area given the difference between the number of births and the number of deaths, which gives the net number of persons added to the population, or the net population increase over the 10 year period
Therefor, the correct option is option d. The area represent the increase in population over a 10-year period
Learn more about finding the area under a curve here:
https://brainly.com/question/19132754
