The population of a region is growing exponentially. There were 40 million people in 1980 (when t=0) and 50 million people in 1990. Find an exponential model for the population (in millions of people) at any time t, in years after 1980.
P(t)=
Predicted population in the year 2000 = _______million people.
What is the doubling time?
Doubling time = _______ years.
Curiously, this is very close to the current population doubling time for India.
To find an exponential model for the population, we can use the general form of an exponential function: P(t) = P₀ * e^(kt), where P(t) is the population at time t, P₀ is the initial population, k is the growth rate constant, and e is Euler's number.
Given that the population in 1980 (t=0) was 40 million (P₀ = 40) and the population in 1990 (t=10) was 50 million, we can substitute these values into the equation to solve for k.
P(t) = P₀ * e^(kt)
50 = 40 * e^(k*10)
Divide both sides by 40:
50/40 = e^(10k)
Take the natural logarithm of both sides to isolate the exponent:
ln(50/40) = 10k
Using a calculator or approximation, we find that ln(50/40) ≈ 0.2231.
Now, solve for k:
10k = ln(50/40)
k = ln(50/40) / 10 ≈ 0.02231
We have found the value of k, so we can now write the exponential model for the population:
P(t) = 40 * e^(0.02231t)
To find the predicted population in the year 2000 (t=20), we substitute t=20 into the equation:
P(t) = 40 * e^(0.02231*20)
P(t) = 40 * e^(0.4462)
Using a calculator or approximation, we find that e^(0.4462) ≈ 1.5627.
P(t) ≈ 40 * 1.5627 ≈ 62.508 million people.
Therefore, the predicted population in the year 2000 is approximately 62.508 million people.
To find the doubling time, we set P(t) = 2P₀ and solve for t:
2P₀ = P₀ * e^(0.02231t)
Divide both sides by P₀:
2 = e^(0.02231t)
Take the natural logarithm of both sides:
ln(2) = 0.02231t
Solve for t:
t = ln(2) / 0.02231 ≈ 31.045 years
Therefore, the doubling time is approximately 31.045 years.
To find an exponential model for the population, we need to use the given information. We know that the population in 1980 (t=0) was 40 million people, and in 1990 (t=10), it was 50 million people.
Step 1: Find the growth rate
To find the growth rate, we can calculate the ratio of the population in 1990 to the population in 1980:
Growth Rate = (Population in 1990 - Population in 1980) / Population in 1980
Growth Rate = (50 million - 40 million) / 40 million
Growth Rate = 10 million / 40 million
Growth Rate = 0.25
Step 2: Write the exponential model equation
The exponential model equation for population growth is given by:
P(t) = P0 * e^(rt)
Where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and e is Euler's number (approximately 2.71828).
In this case, P0 = 40 million and r = 0.25, so the equation becomes:
P(t) = 40 million * e^(0.25t)
Step 3: Find the population in the year 2000
To find the population in the year 2000 (t = 20), substitute t = 20 into the equation:
P(t) = 40 million * e^(0.25 * 20)
P(t) = 40 million * e^5
Using a calculator, we can approximate e^5 ≈ 148.41316. Multiply this value by 40 million:
P(t) ≈ 40 million * 148.41316
P(t) ≈ 5,936.5264 million
Therefore, the predicted population in the year 2000 is approximately 5,936.5264 million (or 5.936 billion) people.
Step 4: Find the doubling time
To find the doubling time, we need to determine how long it takes for the population to double in size. We can use the exponential growth formula to solve for the doubling time.
Doubling Time = ln(2) / r
Where ln is the natural logarithm and r is the growth rate. In this case, r = 0.25, so we have:
Doubling Time = ln(2) / 0.25
Using a calculator, we find that ln(2) ≈ 0.69314718. Substitute this value into the formula:
Doubling Time ≈ 0.69314718 / 0.25
Doubling Time ≈ 2.77258872 years
Therefore, the doubling time is approximately 2.77 years.
Since one of the questions deals with "doubling time", I will use 2 as the base.
(Of course we could use any base, many mathematicians would automatically use e as the base.)
P(t) = a(2)^(kt) , where P(t) is the population in millions, a is the beginning population, and t is the time in years.
clearly , a = 40
P(t) = 40(2)^(kt)
when t=10, (1990), N = 50
50 = 40(2)^(10k)
1.25 = 2^(10k)
take the ln of both sides, hope you remember your log rules
10k = ln 1.25/ln 2
10k = .32193
k = .032193
so P(t) = 40(2)^(.032193t)
in 2000, t = 20
P(20) = 40(2)^(.032193(20))
= 62.5 million
for the formula
P(t) = a(2)^(t/d), d = the doubling time
so changing .032193t to t/d
= .032193t
= t/31.06
so the doubling time is 31.06
another way would be to set
80 = 40(2)^(.032193t)
2 = (2)^(.032193t)
.032193t = ln 2/ln 2 = 1
t = 31.06