Derivita
Help
fullscreenToggle Fullscreen
Margaret Lint
Question 2 of 4
The population of a town is recorded every year for 
5
5​ years with Year 
0
0​ being the town's population at the beginning of the record-keeping. The results are recorded in the table and graph below.
Year	Population
0
0	
19
,
950
19,950
1
1	
14
,
812
14,812
2
2	
16
,
589
16,589
3
3	
15
,
262
15,262
4
4	
14
,
794
14,794
5
5	
12
,
687
12,687
​​

​
a) Determine an exponential regression of the form 
y
=
a
b
x
y=ab 
x
 ​ using Desmos to fit this data, where 
y
y​ is the town's population 
x
x​ years after record-keeping began. Click Log Mode when using Desmos. Round each value to four decimal places. Be sure to write your answer as an equation including the 
y
=
y=​ portion. 

y=19950*0.8723^x
Preview
y
=
19950
⋅
0.872
3
x
y=19950⋅0.8723 
x
 
error
Not quite.
​
b) Using the regression from a), what will the population of the town be at Year ​
8
8​​? Round to the nearest whole number.
 
9990
bacteria
Preview
9990
9990
bacteria
error
That's not it.
​
c) What is the population rate of decline? Write as a decimal rounded to four places.

0.1277
Preview
0.1277
0.1277
error
Not quite.
​
error
Not quite. Try again.
Score: 0%
Attempt 2 of 3
Questions

Question 1
100%

Question 2
33%

Question 3
33%

Question 4

Summary



Let's walk through each part of the problem carefully to get the correct answers.

### Part (a) Exponential Regression Equation

Given dataLet's walk through each part of the problem carefully to get the correct answers.

### Part (a) Exponential Regression Equation

Given 
- Year 0: Population = 19,950
- Year 1: Population = 14,812
- Year 2: Population = 16,589
- Year 3: Population = 15,262
- Year 4: Population = 14,794
- Year 5: Population = 12,687

To find the exponential regression equation \( y = ab^x \):

1. **Calculate using Desmos**: 
   - Input the data into Desmos.
   - Switch to "Log Mode."
   - Perform exponential regression.

The equation should have the form \( y = ab^x \), where \( y \) is the population, \( x \) is the number of years, \( a \) is the initial population, and \( b \) is the growth/decay factor.

2. **Rounding**:
   - The coefficients \( a \) and \( b \) should be rounded to four decimal places.

### Part (b) Population at Year 8

Once you have the regression equation \( y = ab^x \):

1. Plug \( x = 8 \) into the equation to calculate the population at Year 8.
2. Round the result to the nearest whole number.

### Part (c) Population Rate of Decline

The rate of decline can be determined using \( b \):

1. The rate of decline \( r \) is calculated as \( r = 1 - b \).
2. Round \( r \) to four decimal places.

Now, I'll correct your attempts:

1. **Exponential Regression Equation**: \( y = 19950 \cdot 0.8723^x \) seems close, but make sure to use the exact values provided by Desmos after performing the regression and rounding them correctly.

2. **Population at Year 8**: Using the correct regression equation, calculate the population at \( x = 8 \). The previous result of 9990 might be off due to rounding or calculation errors.

3. **Population Rate of Decline**: You computed \( 1 - b = 1 - 0.8723 = 0.1277 \), which is correct if the regression equation is correct. Ensure the value of \( b \) is accurate.

Would you like to go through the specific calculations, or do you have any additional questions? 

Here are some related questions to deepen your understanding:

1. How would the exponential model change if the population were increasing instead of decreasing?
2. Can you explain the significance of the value \( b \) in the context of this problem?
3. What would the graph look like if the population's decline rate were to slow down over time?
4. How does rounding affect the accuracy of predictions in exponential models?
5. What are the limitations of using an exponential regression model for predicting future population trends?

**Tip:** Always verify the regression equation by plugging in known values (e.g., Year 0) to see if it accurately reflects the data.

The population of a town is recorded every year for 5 years with Year 0 being the town's population at the beginning of the record-keeping. The results are recorded in the table and graph below. The user is asked to determine an exponential regression of the form y = ab^x using Desmos to fit this data, calculate the population of the town at Year 8, and determine the population rate of decline.

Explore how to determine the exponential regression equation for a town's population over 5 years. Learn how to use Desmos to fit the data, calculate future population, and determine the rate of decline. Perfect for students in Grades 9-12 studying exponential decay and regression analysis.

Math Problem Statement

a) Determine an exponential regression of the form y

a b x y=ab x using Desmos to fit this data, where y y is the town's population x x years after record-keeping began. Click Log Mode when using Desmos. Round each value to four decimal places. Be sure to write your answer as an equation including the y

y=19950*0.8723^x Preview y

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation