The population of a region is growing exponentially.

ankarskogC

ankarskogC

Answered question

2021-02-20

The population of a region is growing exponentially. There were 10 million people in 1980 (when t=0) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980.
P(t)=?
What population do you predict for the year 2000?
Predicted population in the year 2000 =million people. What is the doubling time?

Answer & Explanation

Nathaniel Kramer

Nathaniel Kramer

Skilled2021-02-21Added 78 answers

Given:
p(0)=10 million ... at t=0, 1980
p(10)=75 million ... at t=10 1990
So, p(0)=p0=10 million
p(t)=p0ekt
p(t)=10ekt
when t=10, (1990) p(10)=75
p(10)=10e10k75=10e10k
e10k=7510
e10k=7.5
Take log on both sides
10k=loge(7.5)
10k=2.02
k=2.0210
k=0.202
k=0.2
p(t)=10e02t
predicted population in year 2000
t=20
p(20)=10e02(20)
p(20)=10e4
p(20)=545.98546
Population in 2000 p(20)=546 million
we, p(t)=10e02t
Now,
The population during doubling time p(t)=2×10 at t=0
p(t)=20
20=100e02t
e02t=2
02t=ln(2)
02t=0.69
t=0.690.2
t=3.47 doubling time

karton

karton

Expert2023-06-19Added 613 answers

We know that in 1980 (when t=0), the population was 10 million people, and in 1990 (when t=10), the population was 75 million people.
We can use the formula for exponential growth:
P(t)=P0×ekt,
where P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate, and t is the time.
Let's solve for the values of P0 and k using the given data:
We have P(0)=10 million and P(10)=75 million.
Plugging in these values into the exponential growth formula:
10=P0×ek×0=P0,
75=P0×ek×10=P0×e10k.
Simplifying the second equation:
e10k=7510=7.5.
Taking the natural logarithm of both sides:
10k=ln(7.5),
k=ln(7.5)10.
Now, we have the value of k. To find P(t) at any time t (in years after 1980), we can use the formula:
P(t)=10×eln(7.5)10×t.
To predict the population in the year 2000 (when t=20), we substitute t=20 into the formula:
P(20)=10×eln(7.5)10×20.
The predicted population in the year 2000 is approximately:
P(20)10×eln(7.5)10×20182.72 million people.
To find the doubling time, we can use the formula:
tdouble=ln(2)ln(7.5)10.
Simplifying this expression:
tdouble=10×ln(2)ln(7.5).
Evaluating this expression:
tdouble4.76 years.
alenahelenash

alenahelenash

Expert2023-06-19Added 556 answers

Step 1: Let's assume that the population of the region at time t years after 1980 is represented by P(t).
Given that the population was 10 million in 1980 (when t=0) and 75 million in 1990, we can use this information to find an exponential model for the population.
Using the general form of an exponential growth function, we have:
P(t)=P0·ekt, where P0 is the initial population and k is the growth rate.
Step 2: Substituting the values for 1980 and 1990, we get the following system of equations:
P(0)=P0·ek·0=10,P(10)=P0·ek·10=75.
Simplifying the equations, we have:
P0=10,P0·e10k=75.
Dividing the second equation by the first equation, we obtain:
e10k=7510=7.5.
To solve for k, we take the natural logarithm (ln) of both sides:
10k=ln(7.5).
Step 3: Finally, we can solve for k:
k=ln(7.5)10.
Thus, the exponential model for the population is:
P(t)=10·eln(7.5)10t.
To predict the population in the year 2000, we substitute t=20 into the model:
P(20)=10·eln(7.5)10·20.
To find the doubling time, we can use the formula for exponential growth:
Doubling Time=ln(2)k, where k is the growth rate we calculated earlier.
Substituting the value of k, we get:
Doubling Time=ln(2)ln(7.5)10.
Simplifying further, we have:
Doubling Time=10ln(2)ln(7.5).
user_27qwe

user_27qwe

Skilled2023-06-19Added 375 answers

Answer:
4.50 years
Explanation:
To find an exponential model for the population, we can use the formula:
P(t)=P0·ekt
where:
- P(t) represents the population at time t (in years after 1980).
- P0 is the initial population at t=0, which is given as 10 million people.
- k is the growth rate constant we need to determine.
- e is Euler's number, approximately equal to 2.71828.
To find the value of k, we can use the information given for the population in 1990:
75=10·ek·10
Solving this equation for k, we divide both sides by 10 and take the natural logarithm:
ln(7510)=10k
Now we can solve for k:
k=ln(7510)10
Using a calculator to evaluate the right-hand side:
k0.154151
Substituting this value back into the exponential model equation, we have:
P(t)=10·e0.154151t
To predict the population in the year 2000, we substitute t=20 into the equation:
P(20)=10·e0.154151·20
Calculating this expression:
P(20)10·e3.0830248.20 million people
Therefore, the predicted population in the year 2000 is approximately 48.20 million people.
Now, let's determine the doubling time, which is the time it takes for the population to double. We can set up the equation:
2P0=P0·ektd
where td represents the doubling time. Dividing both sides by P0 and taking the natural logarithm:
ln(2)=ktd
Solving for td:
td=ln(2)k
Substituting the value of k we found earlier:
td=ln(2)0.154151
Evaluating this expression:
td0.6931470.1541514.50 years
Therefore, the doubling time is approximately 4.50 years.

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