Question

In 2017, the population of a city is 11500, and is increasing exponentially at a rate of 3.1% per year (a) What will the population be in the year 2033? Population size = (b) In what year will the population have tripled? Year = Note You can earn partial credit on this problem

Answer 1

Answer:

(a) The population in the year 2033 is 18630.

(b) After nearly 36years from the starting population of 11500 we will get our population tripled.

Step-by-step explanation:

In 2017, the population of a city is 11500, and is increasing exponentially at a rate of 3.1% per year

This can be formulated with the compound interest formula, also we can think the given situation as we have 11500 rupees which increases at the rate of 3.1% per year.

we can formulate the given situation using the compound interest formula.

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

A is the final population

P is the initial population = 11500

r is the rate at which the population increases = 3.1A =% = 0.031

n denotes the number of times the population increases per year = 1

t is the time or year = 1

A = $11500( 1 + 0.031)$^1*t

(a) The number of years to reach the year 2033 from 2017 is 16years.

so we have the value of t = 16

 so the final population in the year 2033 will be

A = 11500(1+0.031)^1*16

   = 11500(1.031)^16

   = 11500*1.63

A = 18630

Therefore the population size in the year 2033 will be 18630.

(b) We need to find the year in which the population gets tripled. we will triple our population from the population which was in 2017

 so the our equation will be

 11500*3 = 11500(1+0.031)^1*t

  3 = (1.031)^t

upon taking logarithm both sides we get

log(3) = t*log(1.031)    ( using the property log(a^b) = blog(a)

0.477 = t*0.133

t = 0.477/0.0133

t = 35.86years

upon taking approximation we will get 36 years.

Therefore after nearly 36years from the starting population of 11500 we will get our population tripled.