Mathematics, 03.08.2019 01:10 jameanch7182

The birth rate of a population is b(t) = 2100e0.023t people per year and the death rate is d(t)= 1470e0.017t people per year, find the area between these curves for 0 ≤ t ≤ 10. (round your answer to the nearest integer.) what does this area represent? a. this area represents the number of deaths over a 10-year period. b. this area represents the number of births over a 10-year period. c. this area represents the decrease in population over a 10-year period. d. this area represents the increase in population over a 10-year period. e. this area represent the number of children through high school over a 10-year period.

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Answer from: sierravick123owr441

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) = $2,100 \cdot e^{0.023 \cdot t}$

Dirth rate function, d(t) = $1,470\cdot e^{0.017 \cdot t}$

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;

$A = \displaystyle \int\limits^{10}_0 {2,100 \cdot e^{0.023 \cdot t}-1,470\cdot e^{0.017 \cdot t}} \, dt = \left[\frac{2,100}{0.023} \cdot e^{0.023 \cdot t} - \frac{1,470}{0.017} \cdot e^{0.017 \cdot t}\right]^{10}_0$

$\displaystyle A = \left[\frac{2,100}{0.023} \cdot e^{0.023 \cdot t} - \frac{1,470}{0.017} \cdot e^{0.017 \cdot t}\right]^{10}_0 \approx 7,587.9$

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:

link

The birth rate of a population is b(t) = 2100e0.023t people per year and the death rate is d(t)= 147

Answer from: 2kdragginppl

$\boxed{7588; \text{d. the population increase}}$

Step-by-step explanation:

$b(t) = 2100e^{0.023t}\\d(t) = 1470e^{0.017t}$

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

$A = \displaystyle \int_{0}^{10}[b(t) - d(t)]dt\\\\A = \displaystyle \int_{0}^{10}[2100e^{0.023t} - 1470e^{0.017t}]dt\\\\A = \left [ \dfrac{2100}{0.023}e^{0.023t} - \dfrac{1470}{0.017 }e^{0.017t}\right ]_{0}^{10}\\\\$

$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 $\boxed{\mathbf{7588}}$}$

2. Meaning of the area

$\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}$

Answer from: Quest

None of those are technically correct. 9.7 x 104 is actually 1,008.8, but if that answer was a choice, it would be that one. if you meant 9.7 times maybe a thousand or more or divided by 10 or more, than the answer choices would be correct, but other than that, they actually arent.

Answer from: Quest

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