Question

population of a pigeons in a city is 7,000 and is growing exponentially at 19% per
write a function to represent the population of pigeons after t years, where the
terly rate of change can be found from a constant in the function. round all
ficients in the function to four decimal places. also, determine the percentage
of change per quarter, to the nearest hundredth of a percent.

answer attempt 2 out of 3
function: $f(t) = 7000 \left( 1.0441 \
ight)^{4t}$
growth 4.41 % increase per quarter submit answer

Explanation:

Step1: Recall Exponential Growth Formula

The general exponential growth formula is \( P(t)=P_0(1 + r)^n \), where \( P_0 \) is the initial population, \( r \) is the growth rate per period, and \( n \) is the number of periods. For annual growth rate \( R \) (in decimal), to find the quarterly rate \( r \), we know that the annual growth factor is \( 1 + R \), and since there are 4 quarters in a year, \( (1 + r)^4=1 + R \).

Given the annual growth rate \( R = 0.19 \) (19% as a decimal), we solve for \( r \):
\( 1 + r=(1 + 0.19)^{\frac{1}{4}} \)
\( 1 + r = 1.19^{\frac{1}{4}} \)

Step2: Calculate the Quarterly Growth Factor

Calculate \( 1.19^{\frac{1}{4}} \). Using a calculator, \( 1.19^{\frac{1}{4}}=\sqrt[4]{1.19}\approx1.0441 \) (rounded to four decimal places).

Step3: Determine the Quarterly Growth Rate

The growth factor is \( 1 + r = 1.0441 \), so the growth rate \( r=1.0441 - 1=0.0441 \), which is \( 4.41\% \) when converted to a percentage (rounded to the nearest hundredth of a percent).

Step4: Write the Exponential Function

The initial population \( P_0 = 7000 \). The number of quarters in \( t \) years is \( 4t \). So the function is \( f(t)=7000(1.0441)^{4t} \).

Answer:

Function: \( f(t) = 7000(1.0441)^{4t} \)
Percentage of change per quarter: \( 4.41\% \)