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.)
Chapter6: Systems Of Equations And Inequalities
Section6.6: Linear Programming
Problem 40E: Optimal Labor A manufacturer has two different factories that produce three grades of steel:...
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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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