The table in the following figure shows the population of Los Angeles over the years 1930-1990. The accompanying graph shows the corresponding scatter plot and regression line. The equation of the regression line is f(x) = 37,546.068x – 71,238,863.429. y (population) 4 × 10º- 3x 10° Population of Los Angeles Year 2x 10°+ 1930 1,238,048 1,504,277 1,970,358 2,479,015 2,816,061 2,966,850 3,485,398 1940 1950 1 x 10º- 1960 1970 1980 1990 1980 1990 (year) 2000 (a) Use the regression line to compute an estimate for what the population of Los Angeles might have been in 2000. (Round your answer to the nearest thousand.) Compute the percentage error in the estimate, given that the actual figure for 2000 is 3.823 million. (Round your answer to one decimal place.) % (b) Find f-'(x). F(x) = (c) Use your answer in part (b) to estimate the year in which the population of Los Angeles might reach 4.1 million. Hint: For the function f, the inputs are years and the outputs are populations; for f-1, the inputs are populations and the outputs are years. (Round your answer to the nearest year.) 1930 1940 1950 1960 1970
The table in the following figure shows the population of Los Angeles over the years 1930-1990. The accompanying graph shows the corresponding scatter plot and regression line. The equation of the regression line is f(x) = 37,546.068x – 71,238,863.429. y (population) 4 × 10º- 3x 10° Population of Los Angeles Year 2x 10°+ 1930 1,238,048 1,504,277 1,970,358 2,479,015 2,816,061 2,966,850 3,485,398 1940 1950 1 x 10º- 1960 1970 1980 1990 1980 1990 (year) 2000 (a) Use the regression line to compute an estimate for what the population of Los Angeles might have been in 2000. (Round your answer to the nearest thousand.) Compute the percentage error in the estimate, given that the actual figure for 2000 is 3.823 million. (Round your answer to one decimal place.) % (b) Find f-'(x). F(x) = (c) Use your answer in part (b) to estimate the year in which the population of Los Angeles might reach 4.1 million. Hint: For the function f, the inputs are years and the outputs are populations; for f-1, the inputs are populations and the outputs are years. (Round your answer to the nearest year.) 1930 1940 1950 1960 1970
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Population Growth Analysis of Los Angeles (1930-1990)
The table and graph presented outline the population growth of Los Angeles over the span of 60 years, from 1930 to 1990. The trend is modeled using a regression line with the equation:
\[ f(x) = 37,546.068x - 71,238,863.429 \]
#### Graph Description:
- **X-Axis (years)**: Ranges from 1930 to 2000, showing decades.
- **Y-Axis (population)**: Ranges from \(0\) to \(4 \times 10^6\) (or 4 million).
- **Scatter Plot**: Blue dots representing the population at each decade interval (1930, 1940, 1950, 1960, 1970, 1980, 1990).
- **Regression Line**: A red line through the scatter plot showing the linear trend of population growth.
#### Data Table:
| Year | Population of Los Angeles |
|------|----------------------------|
| 1930 | 1,238,048 |
| 1940 | 1,504,277 |
| 1950 | 1,970,358 |
| 1960 | 2,479,015 |
| 1970 | 2,816,061 |
| 1980 | 2,966,850 |
| 1990 | 3,485,398 |
#### Analytical Questions:
**(a) Estimate Population in 2000:**
Using the regression line equation, estimate the population of Los Angeles for the year 2000.
\[ f(2000) = 37,546.068 \times 2000 - 71,238,863.429 \]
**(b) Compute Percentage Error:**
Compare the estimate with the actual population in 2000 (3.823 million). Calculate the percentage error using the formula:
\[ \text{Percentage Error} = \frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\% \]
**(c) Find the Inverse Function \( f^{-1}(x) \):**
Determine the inverse function \( f^{-1}(x) \), where the inputs are populations and the outputs are years.
**(d) Predict Population Reach Year:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ff5dda-0dcd-4838-ba9e-df212171ecd4%2F62a40102-7ede-4843-b08b-b2b7caf853dc%2Ff1xqydv_processed.png&w=3840&q=75)
Transcribed Image Text:### Population Growth Analysis of Los Angeles (1930-1990)
The table and graph presented outline the population growth of Los Angeles over the span of 60 years, from 1930 to 1990. The trend is modeled using a regression line with the equation:
\[ f(x) = 37,546.068x - 71,238,863.429 \]
#### Graph Description:
- **X-Axis (years)**: Ranges from 1930 to 2000, showing decades.
- **Y-Axis (population)**: Ranges from \(0\) to \(4 \times 10^6\) (or 4 million).
- **Scatter Plot**: Blue dots representing the population at each decade interval (1930, 1940, 1950, 1960, 1970, 1980, 1990).
- **Regression Line**: A red line through the scatter plot showing the linear trend of population growth.
#### Data Table:
| Year | Population of Los Angeles |
|------|----------------------------|
| 1930 | 1,238,048 |
| 1940 | 1,504,277 |
| 1950 | 1,970,358 |
| 1960 | 2,479,015 |
| 1970 | 2,816,061 |
| 1980 | 2,966,850 |
| 1990 | 3,485,398 |
#### Analytical Questions:
**(a) Estimate Population in 2000:**
Using the regression line equation, estimate the population of Los Angeles for the year 2000.
\[ f(2000) = 37,546.068 \times 2000 - 71,238,863.429 \]
**(b) Compute Percentage Error:**
Compare the estimate with the actual population in 2000 (3.823 million). Calculate the percentage error using the formula:
\[ \text{Percentage Error} = \frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\% \]
**(c) Find the Inverse Function \( f^{-1}(x) \):**
Determine the inverse function \( f^{-1}(x) \), where the inputs are populations and the outputs are years.
**(d) Predict Population Reach Year:
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