The function f(x) = 4(4)x represents the growth of a fly population every year in a remote swamp. Jackie wants to manipulate the

Question

3 years ago

15

The function f(x) = 4(4)x represents the growth of a fly population every year in a remote swamp. Jackie wants to manipulate the

formula to an equivalent form that calculates three times a year, not just once a year. Which function is correct for Jackie's purpose, and what is the new growth rate?

Mathematics

2 answers:

Yanka [14] · Answer 1 · 2022-07-26T22:53:53+03:00

A growth rate equation has a general form of:

f(x) = P (1 + r)^x

Where,

P = present value

r = rate of growth

x = number of years

If we write the given equation on that form, it would be:

f(x) = 4 (1 + 3)^x

Therefore we can see that the population’s growth rate is 300% per year. To make this formula to an equivalent of three times a year (3x), we have to find the new r.

f(x)_3 = 4 (1 + r)^3x

To calculate this, we must take note that the expression or value of (1 + r)^x must be constant. Therefore equate this with 4^x.

(1 + r)^3x = (4)^x

Taking the logs of both sides:

3x log (1 + r) = x log 4

Cancelling x:

3 log (1 + r) = log 4

3 (1 + r) = 4

1 + r = 4/3

r = 0.33

Therefore the new growth rate is 0.33 or 33%. (ANSWER)

And, the new correct function is: