After its fastest rate of growth ever during the 1980s and 1990s, the rate of growth of world population is expected to slow dramatically in the twenty-first century. The function
G(t) = 1.58e−0.213t
gives the projected annual average percent population growth per decade in the tth decade, with t = 1 corresponding to 2000.†
(a) What will the projected annual average population growth rate be in 2020 (t = 3)? (Round your answer to two decimal places.)
_____% per decade
(b) How fast will the projected annual average population growth rate be changing in 2020 (t = 3)? (Round your answer to two decimal places.)
_____% per decade per decade
To find the projected annual average population growth rate in 2020 (t = 3), we can substitute t = 3 into the function G(t) = 1.58e^(-0.213t).
(a) To calculate this, substitute t = 3 into the function:
G(3) = 1.58e^(-0.213 * 3)
Using a calculator or computer software, evaluate the expression:
G(3) ≈ 1.58e^(-0.639)
The resulting value is approximately 0.5290.
Therefore, the projected annual average population growth rate in 2020 (t = 3) is 0.5290. Converting this to a percentage, we get 52.90%.
So, the answer to part (a) is:
Projected annual average population growth rate in 2020 (t = 3): 52.90% per decade
Now, let's move on to part (b) to find the rate of change of the projected annual average population growth rate in 2020 (t = 3).
To find the rate of change of G(t), we need to calculate its derivative:
G'(t) = (-0.213) * 1.58e^(-0.213t)
For part (b), we need to find the value of G'(3).
G'(3) = (-0.213) * 1.58e^(-0.213 * 3)
Again, using a calculator or computer software, evaluate the expression:
G'(3) ≈ (-0.213) * 1.58e^(-0.639)
The resulting value is approximately -0.2441.
Therefore, the rate of change of the projected annual average population growth rate in 2020 (t = 3) is -0.2441. Converting this to a percentage, we get -24.41%.
So, the answer to part (b) is:
Rate of change of the projected annual average population growth rate in 2020 (t = 3): -24.41% per decade per decade.