Answer:
[tex]f(x)=4(1.32)^{4x}[/tex]
New growth rate = 32%
Step-by-step explanation:
∵ The exponential growth function is,
[tex]f(x)=a(1+b)^x[/tex]
Where,
a = initial value,
b = growth rate per period,
x = number of period,
Here, the given function that shows the population growth every year,
[tex]f(x)=4(3)^x=4(1+2)^x[/tex]
By comparing,
Initial population, a = 4,
Let r be the rate per 4 times a year,
Thus, the function that shows the population growth four times a year,
[tex]f(x)=4(1+r)^{4x}[/tex]
According to the question,
[tex]4(1+r)^{4x}=4(3)^x[/tex]
[tex](1+r)^x=3^x[/tex]
Taking log both sides,
[tex]x\log(1+r)=x\log3[/tex]
[tex]\log(1+r)=\log3[/tex]
By graphing calculator,
[tex]r=0.316=31.6\% \approx 32\%[/tex]
Hence, the required function would be,
[tex]f(x)=4(1+0.32)^x=4(1.32)^x[/tex]
And, rate per period is 32%.
