Question

The population of a parish is growing with an annual percentage rate compounded continuously. The population reaches 1.1 times its previous size in 2 years.

Write the formula to find the annual percentage rate according to the exponential growth function.

Answer 1

Explanation:

The formula for continuous compounding is expressed as

A = Pe(r x t)

Where

A represents the population after t years.

P represents the initial population.

r represents the rate of growth

t represents the time in years.

From the information given,

A = 1.1 × P = 1.1P

t = 2 years

Therefore,

1.1P = Pe(r x 2)

1.1P/P = e(r x 2)

1.1 = e(2r)

Taking ln of both sides of the equation, it becomes

Ln 1.1 = 2rlne

0.095 = 2r

r = 0.095/2

r = 0.0475

Therefore, the formula to find the annual percentage rate is

A = Pe(0.0475t)

Answer 2

4.8%

Explanation:

The population that grows with an annual percentage rate compounded continuously is given by: H = H° e^rt

Let the population be represented by H

The population reaches 1.1 times its previous size in 2 years.

So when t=2,

H = 1.1 H°

We substitute to obtain:

1.1 H° = H° e^r2

1.1 = e^2r

Take natural log to get:

In 1.1 = 2r

r = In 1.1 /2

r = 0.0477

Therefore the annual percentage rate is 4.8%