Question

The population of a town with a 20162016 population of 87 comma 00087,000 grows at a rate of 1.91.9​% per year. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. In what year will the population reach 145 comma 000145,000​?

Answer 1

Answer:

Given,

The initial population, P = 87,000,

Annual rate of increasing, r = 1.9 %,

a) Thus, the population after t years,

$A=P(1+\frac{r}{100})^t$

$A=87000(1+\frac{1.9}{100})^{t}$

$=87000(1+0.019)^t$

$=87000(1.019)^t$

$87000(1.019)^t=87000e^{kt}$

Where, k is the rate constant,

By comparing,

$\implies k=log(1.019)=0.00817418400643\approx 0.00817$

Hence, the approximate value of k is 0.00817.

And, the exponential growth function would be,

$A=87000e^{0.00817t}$

b) If A = 145,000,

$145000=87000(1.019)^t$

By using graphing calculator,

We get,

$t=27.14\approx 27$

The year after 27 years from 2016 is 2043.

Hence, in 2043 the population reach 145,000​.