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) Oh boy, let me crunch some numbers for you. According to the function, G(t) = 1.58e−0.213t, we want to find the projected annual average population growth rate in 2020 (t = 3). So, let's substitute 3 into the equation:

G(3) = 1.58e−0.213(3)

Now, let me grab my calculator... *beep boop beep*

G(3) ≈ 1.58e−0.639 ≈ 0.5305

So, the projected annual average population growth rate in 2020 (t = 3) is approximately 0.5305 percent.

(b) Now, you're asking about how fast the projected annual average population growth rate will be changing in 2020 (t = 3). This means we need to find the derivative of the function G(t) with respect to t. Are you ready for some math acrobatics? Here we go!

G'(t) = -0.213(1.58)e−0.213t

Now, let's substitute t = 3 and evaluate the derivative at that point:

G'(3) = -0.213(1.58)e−0.213(3)

*calculating intensifies*

G'(3) ≈ -0.213(1.58)e−0.639 ≈ -0.131

So, the projected annual average population growth rate will be changing at a rate of approximately -0.131 percent per decade per decade in 2020 (t = 3). Whoa, that's quite a change, isn't it?

(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) = 1.58e^-0.213t.

G(3) = 1.58e^-0.213(3)
= 1.58e^-0.639
≈ 1.58 * 0.528
≈ 0.83424

Therefore, the projected annual average population growth rate in 2020 is approximately 0.83% per decade.

(b) The rate of change of the projected annual average population growth rate can be found by taking the derivative of G(t) with respect to t, and then substituting t = 3.

G'(t) = -0.213 * 1.58e^-0.213t

G'(3) = -0.213 * 1.58e^-0.213(3)
= -0.213 * 1.58e^-0.639
≈ -0.285 * 0.528
≈ -0.15048

Therefore, the rate of change of the projected annual average population growth rate in 2020 is approximately -0.15% per decade per decade.

To find the projected annual average population growth rate in 2020 (t = 3), we need to substitute the value of t into the function G(t) = 1.58e^(-0.213t).

(a) Substituting t = 3 into the equation, we get:
G(3) = 1.58e^(-0.213 * 3)

Calculating this expression:

G(3) ≈ 1.58e^(-0.639) ≈ 0.635

So, the projected annual average population growth rate in 2020 (t = 3) is approximately 0.635%. Rounded to two decimal places, it is 0.64%.

To find the rate at which 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) Differentiating the function G(t) = 1.58e^(-0.213t) with respect to t:

G'(t) = -0.213 * 1.58e^(-0.213t)

Substituting t = 3 into the derivative expression:

G'(3) = -0.213 * 1.58e^(-0.213 * 3)

Calculating this expression:

G'(3) ≈ -0.213 * 1.58e^(-0.639) ≈ -0.151

So, the projected annual average population growth rate is changing at approximately -0.151% per decade in 2020 (t = 3). Rounded to two decimal places, it is -0.15%.