According to the US Census, the world population P, in billions, is approximately P=6.342e^(0.011t); where t is in years since January 1, 2004. At what rate was the world's population increasing on January 1, 2005? Give your answer in millions per year.
Is the answer 6412 millions per year?
To find the rate at which the world's population was increasing on January 1, 2005, we need to calculate the derivative of the population equation with respect to time, and then evaluate it for t = 1 (since 1 year has passed since January 1, 2004).
First, let's find the derivative of the population equation:
P = 6.342e^(0.011t)
Taking the derivative with respect to t:
dP/dt = (0.011)(6.342)e^(0.011t)
Now, let's evaluate this derivative at t = 1:
dP/dt = (0.011)(6.342)e^(0.011 * 1)
= 0.069762
The rate at which the world's population was increasing on January 1, 2005, is approximately 0.069762 billion per year. To convert this to millions per year, we multiply by 1000:
0.069762 billion/year * 1000 million/billion = 69.762 million/year
Therefore, the correct answer is 69.762 million per year, not 6412 million per year.
To find the rate at which the world's population was increasing on January 1, 2005, we need to calculate the derivative of the population function with respect to time.
The population function is given as P = 6.342e^(0.011t).
To find the derivative, we differentiate the function using the chain rule. The derivative of e^(0.011t) is 0.011e^(0.011t).
Therefore, the derivative of the population function, dP/dt, is 6.342 * 0.011e^(0.011t).
To find the rate of population increase on January 1, 2005, we substitute t = 1 into the derivative function.
dP/dt = 6.342 * 0.011e^(0.011 * 1)
dP/dt = 0.069762e^(0.011)
Calculating this expression, we get:
dP/dt ≈ 0.069762(1.011) ≈ 0.070504 million per year
Therefore, the rate of population increase on January 1, 2005, was approximately 0.070504 million per year.
So, the answer of 6412 millions per year is incorrect. The correct answer is approximately 0.070504 million per year.
Did you even consider if your answer makes any sense ??
6412 million is 6.4 billion.
Do you really think that the earth's population increased at a rate of 6.4 billion per year if the entire population is 6.3 billion??
According to your formula
dP/dt = .011(6.342)e^.011t
so when t=1 (from 2004 to 2005)
dP/dt = 70.53 million/year