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
(a) In order to find the projected annual average population growth rate in 2020 (t = 3), we need to plug t=3 into the function G(t):
G(3) = 1.58e^(-0.213*3)
Using a calculator, we can evaluate this expression:
G(3) ≈ 1.58e^(-0.639)
G(3) ≈ 1.58 * 0.527
G(3) ≈ 0.83
Therefore, the projected annual average population growth rate in 2020 is approximately 0.83% per decade.
(b) To find how fast the projected annual average population growth rate is changing in 2020 (t = 3), we need to find the derivative of the function G(t) with respect to t:
G'(t) = -0.213 * 1.58e^(-0.213t)
Now, we can plug t=3 into G'(t):
G'(3) = -0.213 * 1.58e^(-0.213*3)
Using a calculator, we can evaluate this expression:
G'(3) ≈ -0.213 * 1.58e^(-0.639)
G'(3) ≈ -0.428 * 0.527
G'(3) ≈ -0.225
Therefore, the projected annual average population growth rate is changing at approximately -0.225% per decade per decade in 2020.
(a) To find the projected annual average population growth rate in 2020 (t = 3), we need to substitute t = 3 into the given function G(t).
G(3) = 1.58e^(-0.213 * 3)
Using a calculator, we can evaluate this expression.
G(3) ≈ 1.58e^(-0.639)
G(3) ≈ 1.58 * 0.527
G(3) ≈ 0.83166
The projected annual average population growth rate in 2020 is approximately 0.83% per decade.
(b) To find how fast the projected annual average population growth rate is changing in 2020 (t = 3), we need to find the derivative of the function G(t) with respect to t.
G'(t) = -0.213 * 1.58e^(-0.213t)
G'(3) = -0.213 * 1.58e^(-0.213 * 3)
Using a calculator, we can evaluate this expression.
G'(3) ≈ -0.213 * 1.58e^(-0.639)
G'(3) ≈ -0.213 * 0.527
G'(3) ≈ -0.11265
The projected annual average population growth rate is changing at a rate of approximately -0.11% per decade per decade in 2020.
To find the projected annual average population growth rate in 2020 (t = 3), we need to substitute t = 3 into the function G(t) = 1.58e^(-0.213t).
(a) Plugging in t = 3:
G(3) = 1.58e^(-0.213 * 3)
G(3) = 1.58e^(-0.639)
Using a calculator or computer program, we can evaluate this expression to find the answer.
The projected annual average population growth rate in 2020 will be x% per decade (rounding to two decimal places).
To find how fast the projected annual average population growth rate is changing in 2020 (t = 3), we need to find the derivative of the function G(t) with respect to t.
(b) Taking the derivative of G(t) = 1.58e^(-0.213t) with respect to t:
G'(t) = -0.213 * 1.58 * e^(-0.213t)
Plugging in t = 3:
G'(3) = -0.213 * 1.58 * e^(-0.213 * 3)
G'(3) = -0.213 * 1.58 * e^(-0.639)
Again, using a calculator or computer program, we can evaluate this expression to find the answer.
The rate of change of the projected annual average population growth rate in 2020 will be y% per decade per decade (rounding to two decimal places).
a. G(t0 = 1.58*e^n.
n = -0.213*t = -0.213*3 = -0.639.
G(t) = 1.58*e^-0.639=1.58*0.5278=0.83.
% per decade.
b. 0.83%/dec. * (1/10dee.) =0.08 % per
decade per decade.