1.604

QUESTION 14
Solve.
The population of a particular country was 23 million in 1981; in 1987, it was 32 million. The exponential growth function $\mathrm{A}=23 \mathrm{e}^{\mathrm{kt}}$ describes the population of this country $t$ years after 1981. Use the fact that 6 years after 1981 the population increased by 9 million to find $\mathrm{k}$ to three decimal places.

Step 1 ：The population of a particular country was 23 million in 1981; in 1987, it was 32 million. The exponential growth function \(A=23e^{kt}\) describes the population of this country \(t\) years after 1981. We are asked to use the fact that 6 years after 1981 the population increased by 9 million to find \(k\) to three decimal places.

Step 2 ：First, we substitute the given values into the exponential growth function. In 1987, which is 6 years after 1981, the population \(A\) was 32 million. So we have \(32 = 23e^{6k}\).

Step 3 ：We can solve this equation for \(k\) by first dividing both sides by 23 to get \(32/23 = e^{6k}\).

Step 4 ：Then, we take the natural logarithm of both sides to get \(ln(32/23) = 6k\).

Step 5 ：Finally, we divide both sides by 6 to find \(k = ln(32/23) / 6\).

Step 6 ：Calculating this gives \(k \approx 0.055\).

Step 7 ：Final Answer: The growth rate \(k\) to three decimal places is \(\boxed{0.055}\).