In 2010 the birthrate in a certain country increases from 11*(e^(0.04*t)) million births per year to 16*(e^(0.04*t)) million births per year, where t is the number of years since January 1, 2010.
Set up the integral and find the projected increase in population that would result due to the higher birth rate between January 1, 2010 and January 1, 2029.
(Express the limits as exact numbers.)
Not sure what difference it makes what the rate was before 2010, since we are told that the rate after 2010 is 16e^(.04t) per year after 2010.
So, now we have
dp/dt = 16e^(.04t)
so, during the next 19 years, the population grows by
∫[0,19] 16e^(.04t) dt
To find the projected increase in population due to the higher birth rate between January 1, 2010, and January 1, 2029, we need to calculate the integral of the birth rate function over that time period.
The integral represents the accumulation of the birth rate over time and can be calculated as follows:
∫[January 1, 2010 to January 1, 2029] (16 * e^(0.04t) - 11 * e^(0.04t)) dt
To evaluate this integral, we need to determine the limits of integration. In this case, the birth rate is changing from January 1, 2010, to January 1, 2029, which is a span of 19 years.
Therefore, the lower limit of integration is 0 (representing January 1, 2010), and the upper limit of integration is 19 (representing January 1, 2029).
Now we can rewrite the integral with the specific limits:
∫[0 to 19] (16 * e^(0.04t) - 11 * e^(0.04t)) dt
To evaluate this integral, we can use the properties of integrals and the rules of differentiation. Integration of e^(0.04t) results in (1/0.04) * e^(0.04t), and integration of constants follows the standard rules.
∫[0 to 19] (16 * e^(0.04t) - 11 * e^(0.04t)) dt
= ∫[0 to 19] (16/0.04 * e^(0.04t) - 11/0.04 * e^(0.04t)) dt
Simplifying further:
= ∫[0 to 19] (400 * e^(0.04t) - 275 * e^(0.04t)) dt
= ∫[0 to 19] (125 * e^(0.04t)) dt
Now, integrating 125 * e^(0.04t) gives us:
= (125/0.04) * e^(0.04t) evaluated from 0 to 19
= 3125 * (e^(0.04*19) - e^(0.04*0))
Simplifying further:
= 3125 * (e^(0.76) - 1)
Now we substitute the numerical value of e^(0.76) into the equation:
≈ 3125 * (2.137537 - 1)
≈ 3125 * 1.137537
≈ 3554.1789 million births
Therefore, the projected increase in population due to the higher birth rate between January 1, 2010, and January 1, 2029, is approximately 3554.1789 million births.