Question

graded assignment
unit test, part 2: exponential and logarithmic functions
answer the questions below. when you are finished, submit this test to your teacher for full credit.
total score: ____ of 15 points

(score for question 1: ___ of 5 points)

1. the population of a town was 5655 in 2010. the population grows at a rate of 1.4% annually.

(a) use the exponential growth model to write an equation that estimates the population t years after 2010.
(a) estimate the population of the town in 2022. show your work.
answer:

Explanation:

Step1: Write the exponential growth formula

The exponential growth formula is \( P(t)=P_0(1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate as a decimal, and \( t \) is the number of years. Given \( P_0 = 5655\), \( r=0.014\). So the equation is \( P(t)=5655(1 + 0.014)^t=5655(1.014)^t \).

Step2: Calculate the value of \( t \) for 2022

The year 2022 - 2010 = 12 years. So \( t = 12 \).

Step3: Substitute \( t = 12 \) into the formula

\( P(12)=5655\times(1.014)^{12}\).
First, calculate \( (1.014)^{12}\approx1.1802 \) (using a calculator).
Then \( P(12)=5655\times1.1802\approx6675 \).

Answer:

(a) The equation is \( P(t) = 5655(1.014)^t \). (b) The population in 2022 is approximately \( 6675 \).