Question

exponential regression
the table below shows the population of a fictional california gold rush town named lehi in the years after its peak population in 1880.

year	1880	1890	1900	1910	1920	1930
population	8900	5119	4839	4389	2631	1649

for the purpose of this problem, let p represent the population of lehi t years after 1880 (t = 0 represents 1880). the new table is:

t	0	10	20	30	40	50
p(t)	8900	5119	4839	4389	2631	1649

use your calculator to determine the exponential regression equation that models the set of data above. round the \a\ value to two decimals, and round the \b\ value to three decimals. use the indicated variables and proper function notation.
p(t)=
based on the your regression model, what is the percent decrease per year?
%
find p(55). round your answer to the nearest whole number.
p(55)=
interpret your answer by completing the following sentence. be sure to use units in your answer.
\the population of lehi after 1880 was about .\
how long did it take for the population of lehi to reach 390 people? round your answer to the nearest whole number.
p(t)=390 when t =
interpret your answer by completing the following sentence. be sure to use units in your answer.
in after 1880, the population of lehi was about .
how long did it take for the population of lehi to drop by half? round your answer to the nearest whole number.
p(t) has halved when t =
interpret your answer by completing the following sentence. be sure to use units in your answer.
in after 1880, the population of lehi had dropped by about half.

Explanation:

Step1: Enter data into calculator

Enter the pairs of values \((t, P(t))\) where \(t = 0,10,20,30,40,50\) and \(P(t)\) are the corresponding population values into a graphing - calculator's regression feature. The general form of an exponential regression equation is \(P(t)=ab^{t}\).

Step2: Find the regression equation

Using a calculator, the exponential regression equation is \(P(t)=8869.77\times0.983^{t}\) (rounded as required).

Step3: Find the percent decrease per year

The general form of an exponential decay function is \(y = a(1 - r)^{x}\), where \(r\) is the rate of decay. In the equation \(P(t)=ab^{t}\), \(b = 1 - r\). So, \(r=1 - b\). Here, \(b = 0.983\), so \(r=1 - 0.983=0.017\) or \(1.7\%\).

Step4: Find \(P(55)\)

Substitute \(t = 55\) into \(P(t)=8869.77\times0.983^{t}\).
\[P(55)=8869.77\times0.983^{55}\]
\[P(55)=8869.77\times0.3977\]
\[P(55)\approx3528\]

Step5: Find \(t\) when \(P(t)=390\)

Set \(P(t)=390\) in the equation \(P(t)=8869.77\times0.983^{t}\).
\[390 = 8869.77\times0.983^{t}\]
\[\frac{390}{8869.77}=0.983^{t}\]
Take the natural - logarithm of both sides: \(\ln(\frac{390}{8869.77})=t\ln(0.983)\)
\[t=\frac{\ln(\frac{390}{8869.77})}{\ln(0.983)}\]
\[t=\frac{\ln(390)-\ln(8869.77)}{\ln(0.983)}\]
\[t\approx137\]

Step6: Find \(t\) when \(P(t)\) is halved

The initial population \(P(0)=8869.77\), and we want to find \(t\) when \(P(t)=\frac{8869.77}{2}=4434.885\).
Set \(P(t)=4434.885\) in \(P(t)=8869.77\times0.983^{t}\).
\[4434.885 = 8869.77\times0.983^{t}\]
\[\frac{4434.885}{8869.77}=0.983^{t}\]
Take the natural - logarithm of both sides: \(\ln(\frac{4434.885}{8869.77})=t\ln(0.983)\)
\[t=\frac{\ln(\frac{4434.885}{8869.77})}{\ln(0.983)}\]
\[t=\frac{\ln(4434.885)-\ln(8869.77)}{\ln(0.983)}\]
\[t\approx40\]

Answer:

1. \(P(t)=8869.77\times0.983^{t}\)
2. \(1.7\)
3. \(3528\)
4. The population of Lehi \(55\) years after 1880 was about \(3528\) people.
5. \(137\)
6. In \(137\) years after 1880, the population of Lehi was about \(390\) people.
7. \(40\)
8. In \(40\) years after 1880, the population of Lehi had dropped by about half.