for population in billions, where t is years since 1990:
N(t) = 41/1+11 e^−0.1 t
(b) According to this model, when will the earth's population reach 20.5 billion?
During the year .
How about 40.9 billion?
During the year .
To find the year when the Earth's population will reach a certain number, we can use the given population model and solve for the value of t.
For the first case, when the population reaches 20.5 billion, we can set N(t) equal to 20.5 and solve for t:
20.5 = (41/(1+11)) * e^(-0.1t)
To isolate e^(-0.1t), we can multiply both sides of the equation by (1+11)/41:
((1+11)/41) * 20.5 = e^(-0.1t)
Now, take the natural logarithm (ln) of both sides to remove the exponent:
ln(((1+11)/41) * 20.5) = -0.1t
Simplify the left side of the equation:
ln(12/41 * 20.5) = -0.1t
Using a calculator, we find that ln(12/41 * 20.5) is approximately -1.112.
Now, divide both sides of the equation by -0.1:
(-1.112)/(-0.1) = t
This gives us t ≈ 11.12 years.
Since t represents the number of years since 1990, to find the year when the Earth's population will reach 20.5 billion, we add 11.12 years to 1990:
1990 + 11.12 ≈ 2001.12
Therefore, the Earth's population will reach 20.5 billion during the year 2001.
Now let's solve for the second case, when the population reaches 40.9 billion, following the same steps as above:
40.9 = (41/(1+11)) * e^(-0.1t)
((1+11)/41) * 40.9 = e^(-0.1t)
ln(((1+11)/41) * 40.9) = -0.1t
Using a calculator, ln(12/41 * 40.9) is approximately 0.684.
(-0.684)/(-0.1) = t
This gives us t ≈ 6.84 years.
Adding 6.84 years to 1990:
1990 + 6.84 ≈ 1996.84
Therefore, the Earth's population will reach 40.9 billion during the year 1996.
To summarize:
- The Earth's population will reach 20.5 billion during the year 2001.
- The Earth's population will reach 40.9 billion during the year 1996.