Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table and the method of this example to find an equation that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute). Weight, x Galloping Speed, y 25 193.5 35 50 185.7 174.8 25 193.5 75 500 1000 162.2 124.9 111.2 Take the natural logarithm of each coordinate to obtain points of the form (In x, In y). (Round your answers to three decimal places.) Weight, x 500 35 185.7 50 174.8 124.9 Galloping Speed, y In x In y 75 162.2 1000 111.2 Find the least squares regression line for the transformed points. (Round your answers to three decimal places.) In y Tin x 1-1 Find an equation of the form y = ax that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute). (Round your value for a to one decimal place and your value for b to three decimal places.)
Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table and the method of this example to find an equation that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute). Weight, x Galloping Speed, y 25 193.5 35 50 185.7 174.8 25 193.5 75 500 1000 162.2 124.9 111.2 Take the natural logarithm of each coordinate to obtain points of the form (In x, In y). (Round your answers to three decimal places.) Weight, x 500 35 185.7 50 174.8 124.9 Galloping Speed, y In x In y 75 162.2 1000 111.2 Find the least squares regression line for the transformed points. (Round your answers to three decimal places.) In y Tin x 1-1 Find an equation of the form y = ax that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute). (Round your value for a to one decimal place and your value for b to three decimal places.)
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter1: Equations And Graphs
Section1.10: Modeling Variation
Problem 48E
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Transcribed Image Text:**Educational Content: Analyzing Galloping Speed in Relation to Animal Weight**
Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table and method in this example to find an equation that relates an animal's weight \( x \) (in pounds) and its lowest galloping speed \( y \) (in strides per minute).
**Data Table:**
| Weight, \( x \) | 25 | 35 | 50 | 75 | 500 | 1000 |
|-------------------|------|-------|-------|-------|-------|-------|
| Galloping Speed, \( y \) | 193.5 | 185.7 | 174.8 | 162.2 | 124.9 | 111.2 |
**Instructions:**
1. **Transform Data Using Natural Logarithms:**
- Convert each coordinate to points of the form \((\ln x, \ln y)\).
- Round your answers to three decimal places.
**Transformed Data:**
| Weight, \( x \) | 25 | 35 | 50 | 75 | 500 | 1000 |
|-------------------|----------|----------|----------|----------|----------|----------|
| Galloping Speed, \( y \) | 193.5 | 185.7 | 174.8 | 162.2 | 124.9 | 111.2 |
| \(\ln x\) | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ |
| \(\ln y\) | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ | \_\_\_\_ |
2. **Calculate the Least Squares Regression Line:**
- Use the transformed
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