MATH SOLVE

2 months ago

Q:
# 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 β€ t β€ 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.

Accepted Solution

A:

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]