Question

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?
Answer Options:

f(x) = 4(4)^x; growth rate 400%

f(x) = 4(4)^3x; growth rate 4%

f(x) = 4(1.59)^3x; growth rate 59%

f(x) = 4(1.59)^x; growth rate 4%

Answer 1

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

calculates three times a year, not just once a year.

$f(x)= 4(4)^x$

3 times a year

so x becomes 3x

$f(x)= 4(1+r)^{3x}$

$4(4)^x=4(1+r)^{3x}$

Take log on both sides

$log(4)^x=log(1+r)^{3x}$

$log(4)^x=log(1+r)^{3x}$

Use log property and move exponent before log

$xlog(4)=3xlog(1+r)}$

Divide both sides by x

log 4 = 3 log(1+r)

Solve for '1+r'

log 4 = log(1+r)^3

Remove log from both sides

4 = (1+r)^3

take cube root on both sides

1.584740= 1+r

1+r = 1.59

$f(x)= 4(1+r)^{3x}$

so equation becomes

$f(x)= 4(1.59)^{3x}$

1+r = 1.59

subtract 1 from both sides

So r= 59 = 59%

So growth factor is 59%

Answer is option C