SOLUTION: The population of Plano, Texas, which follows the exponential growth model, increased from 222,030 in 2000 to 259,841 in 2010. (Source: U.S. Census Bureau, www.census.gov) Wor

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The population of Plano, Texas, which follows the exponential growth model, increased from 222,030 in 2000 to 259,841 in 2010. (Source: U.S. Census Bureau, www.census.gov) Wor      Log On


   

Question 1157957: The population of Plano, Texas, which follows the exponential growth model, increased from
222,030 in 2000 to 259,841 in 2010. (Source: U.S. Census Bureau, www.census.gov)
Work is required!
a. Find the exponential growth rate, k. Give the exact rate.

b. Write the exponential growth function that models the population after t years.
c. What is the projected population in 2020?
d. How long should it take the population to double?
Answer by KMST(5396) About Me  (Show Source):
You can put this solution on YOUR website!
Exponential functions are functions where the variable is in the exponent, such as y=2%5Ex .
Exponential functions of the form y=A%2Ae%5Ekt where A and k are constants
are used to model exponential growth and exponential decay.
Let's call our function P for population.
Let t be the number of years after 2000.
The function is P=A%2Ae%5Ekt .
For the year 2000, t=0 and P=222030 , and
222030=A%2Ae%5E%28k%2A0%29 --> 222030=A%2Ae%5E0 --> 222030=A%2A1%7D%7D+--%3E%7B%7B%7Bhighlight%28A=222030%29
For the year 2010, t=10 , P=259841 , and
259841=222030%2Ae%5E%28k%2A10%29 --> 259841%2F222030=e%5E%28k%2A10%29 --> ln%28259841%2F222030%29=k%2A10 --> highlight%28k=ln%28259841%2F222030%29%2F10%29
That is the exact value.
k=approximately0.0157257 or k=approximately0.01573 .

We could write the growth function as
P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29 , or
P=222030%2Ae%5E%28ln%28%22259841%2F222030%22%29%2A%28%22t%2F10%22%29%29 , or
P=222030%2A%28e%5Eln%28%22259841%2F222030%22%29%29%5E%28%22t%2F10%22%29 , or P=222030%2A%28259841%2F222030%29%5E%28%22t%2F10%22%29 ,
or maybe use the approximate value for k, and write it as
P=222030%2Ae%5E%280.01573t%29

We can calculate the projected population in 2010 and 2020, using the equations found above.
For 2010, t=10 . Substituting that value into P=222030%2Ae%5E%280.01573t%29 , we get
P=222030%2Ae%5E%280.01573%2A10%29=222030%2Ae%5E0.1573=222030%2A1.17034667=259852(rounded to whole number).
However using P=222030%2Ae%5E%280.0157257t%29 , we get
P=222030%2Ae%5E%280.0157257%2A10%29=222030%2Ae%5E0.157257=222030%2A1.17029634=25984.08967%22=%22259841(rounded to whole number).
To match all 6 digits in the population for 2010, we need the exact value of k , or at least a better approximation and we need to carry more digits through the calculations.
For 2020 t=20 , substituting it into P=222030%2Ae%5E%280.0157257t%29 we get
.
Rounding to whole numbers, we get that the projected population in 2020 is highlight%28304091%29 .
If we use the more accurate P=222030%2A%28259841%2F222030%29%5E%22t%2F10%22 , we get
.
Rounding to whole numbers, we get that the projected population in 2020 is highlight%28304091%29 .

Using the equation with the exact value of k , P=222030%2Ae%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29 , we could set P=2%2A222030=444060 and solve for t .
We get
2=e%5E%28%28%22ln%28259841%2F222030%29%2F10%22%29%2At%29-->ln%282%29=%22ln%28259841%2F222030%29%2F10%22%2At-->t=10%2Aln%282%29%2F%22ln%28259841%2F222030%29%22=44.077 (rounded to 3 decimal places).
Rounding to whole numbers, it takes highlight%2844%29 years for the population to double at the rate observed between 2000 and 2010.