Assuming the population of the earth changes at a rate proportional to the current population further, it is estimated that at time t=0, the earth's population was 600 million, at t=300, it's population was 2.8 billion. find an expression giving the population of the earth at anytime. assuming that the greatest population the earth can support is 25 billion, when will this limit be reached?
population changes at a rate proportional to the current population:
dp/dt = kp
dp/p = k dt
ln p = kt+c
p = c*e^(kt)
Let p be measured in units of millions. Then we have
p(0) = 600
p(300) = 2800
so,
p(t) = 600e^(kt)
600e^(300k) = 2800
e^(300k) = 14/3
300k = ln(14/3)
k = ln(14/3)/300 = 0.00513
p(t) = 600 e^(0.00513t)
To find an expression giving the population of the Earth at any time, we can use the formula for exponential growth. Let's assume that the population at time t is P(t).
We know that the population changes at a rate proportional to the current population. This can be represented by the differential equation:
dP/dt = k * P
Where:
- dP/dt is the rate of change of population with respect to time
- k is the proportionality constant
To find the value of k, we can use the population values given in the problem. At time t=0, the population was 600 million (600,000,000), and at t=300, the population was 2.8 billion (2,800,000,000). Substituting these values into the differential equation, we get:
dP/dt = k * P
600,000,000 * k = 2,800,000,000
Solving for k, we find:
k = 2,800,000,000 / 600,000,000
k ≈ 4.67
Now we have the constant k. We can rewrite the differential equation as:
dP/dt = 4.67 * P
This is a separable differential equation, which can be solved by separating variables and integrating:
(1 / P) dP = 4.67 dt
Integrating both sides:
∫ (1 / P) dP = ∫ 4.67 dt
ln|P| = 4.67t + C
Where ln|P| is the natural logarithm of the absolute value of P, and C is the constant of integration.
Exponentiating both sides:
|P| = e^(4.67t+C)
Since e^C is just a constant, we can rewrite it as another constant A:
|P| = A * e^(4.67t)
Now, we know that the greatest population the Earth can support is 25 billion (25,000,000,000). So we need to find the value of A that satisfies this condition.
When the Earth's population reaches this limit, we have:
|P| = 25,000,000,000
Substituting this into the expression for P, we get:
25,000,000,000 = A * e^(4.67t)
To find when this limit will be reached, we can solve for t. Taking the natural logarithm of both sides:
ln|25,000,000,000| = ln|A * e^(4.67t)|
ln|25,000,000,000 / A| = 4.67t
Solving for t:
t = ln|25,000,000,000 / A| / 4.67
Now you can calculate the value of t by substituting the appropriate values for A (the constant from before) and solving the equation.