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