Question

The birth rate in a certain city is described by the following function b(t)=5.48 -0.01 t^2. The city's death rate is given by d(t)=5+0.02 t^2. Here, t is measured in years, and t = 0 corresponds to the start of the year 1990. The birth and death rates are measured in thousands of births or deaths per year. At the start of 1990, the population of the city is 150 thousand. Enter all the following answers correct to two decimal places.

Calculate the total number of births between the start of 1990 and the end of 1999. Number of births = ____ thousand.
Calculate the total number of deaths over the same period Number of deaths = ____ thousand.
What is the population of the city at the start of year 2000? Population : _____ thousand.
Considering just the period from the start of 1990 to the start of 2000, over what interval is the population increasing? (Enter your dates as years, not the number of years since 1990.)
Over what interval is the population decreasing?
Calculate the area between the curves y = b ( t ) and y = d ( t ) for t between 0 and 10. Area :

Answer 1

Answer:

1. 4,67 thousand

2. 6,62 thousand

3. 148,05 thousand

4. The population increases from 1991 to 1994, and decreases from 1995 to 1999

5. Area between the curves:

y=b(t)=51,47

y=d(t)=56,67

Step-by-step explanation:

1.Total number of births in 1999. Evaluating the function b(t) at t=1999-1990=9

$b(9)=5,48-0,01(9)^{2} =4,67$

2. Total number of births in 1999. Evaluating the function d(t) at t=1999-1990=9

$d(9)=5+0,02(9)^{2} =6,67$

3. The equation that calculate the population of the city in certain period of time is:

$P(t)=150+b(t)-d(t)$

In this case, the population at the star of the year 2000 could be calculate by evaluating the function at t=9. Then, replacing the results of b(9) and d(9):

$P(9)=150+1,67-6,62=148,05$

4. Evaluate the function $P(t)=150+b(t)-d(t)$ at t=0,1,2,...,9 (See the attachment)

The values show that the population increases during 1991 and 1994, and decreases during 1995 and 1999.

5. Calculating the area between the curves requires to calculate the integrated and evaluated at the interval given:

For y=b(t):

$y=b(t)=5,48-0,01t^{2}\\ y=b(t)=\int\limits^0_10 {5,48-0,01t^{2}} \, dt\\y=b(t)=5,48t-(0,01/3)t^{3}$

Evaluating the function at t=0 and t=10:

$b(10)-b(0)=5.48(10)-(0,01/3)(10)^3-5.48(0)+(0,01/3)(0)^3\\b(10)-b(0)= 51,47$

• For y=d(t):

$y=d(t)=5+0,02t^{2}\\ y=d(t)=\int\limits^0_10 {5+0,02t^{2}} \, dt\\y=d(t)=5t+(0,02/3)t^{3}$

Evaluating the function at t=0 and t=10:

$d(10)-b(0)=5(10)-(0,02/3)(10)^3-5(0)-(0,02/3)(0)^3\\d(10)-d(0)= 56,67$

Answer 2

a) There are 51467 births between the start of 1990 and the end of 1999.

b) There are 56667 death between the start of 1990 and the end of 1999.

c) The population of the city at the start of year 2000 is 144800.

d) The population is increasing in the period 1990 - 1993.

e) The population is decreasing in the period 1994-1999.

f) The area between the curves has a total value of -5200.

a) The total number of births is the definite integral of the birth rate function, that is to say:

$B = \int\limits_{0}^{10} b(t)\,dt = 5.48\int\limits_{0}^{10} dt - 0.01\int\limits_{0}^{10}t^{2} \,dt$

$B = 5.48\cdot t|_{0}^{10}-0.01\cdot \left(\frac{x^{3}}{3} \right)|_{0}^{10}$

$B = 51.467$

There are 51467 births between the start of 1990 and the end of 1999.

b) The total number of deaths is the definite integral of the death rate function, that is to say:

$D = \int\limits_{0}^{10} d(t)\,dt = 5\int\limits_{0}^{10}dt + 0.02\int\limits_{0}^{10}t^{2}\,dt$

$D = 5\cdot t|_{0}^{10} + 0.02\cdot \left(\frac{x^{3}}{3} \right)|_{0}^{10}$

$D = 56.667$

There are 56667 death between the start of 1990 and the end of 1999.

c) The population of the city at the start of year 2000 is equal to the population of the city at the start of 1990 plus the total number of births and minus the total number of deaths:

$P_{2000} = P_{1990} + B - D$ (1)

If we know that $P_{1990} = 150000$, $B = 51467$ and $D = 56667$, then the population of the city at the start of year 2000 is:

$P_{2000} = 150000+51467-56667$

$P_{2000} = 144800$

The population of the city at the start of year 2000 is 144800.

d) With the help of graphing tool and based on the fact that city is growing when birth rates are greater than death rates, we conclude that population is increasing in the period 1990 - 1993.

e) With the help of graphing tool and based on the fact that city is decreasing when birth rates are less than death rates, we conclude that population is decreasing in the period 1994-1999.

f) The area between the curves is determined by the following definite integral:

$A = \int \limits_{0}^{10} [b(t)-d(t)]\,dt = \int\limits_{0}^{10}(0.48-0.03\cdot t^{2})\,dt$

$A = 0.48\cdot t|_{0}^{10}-0.03\cdot \left(\frac{t^{3}}{3} \right)|_{0}^{10}$

$A = -5.2$

The area between the curves has a total value of -5200.

We kindly invite to check this question on definite integrals: https://brainly.com/question/9897385