17. A robot begins moving on a linear path at (3, 1). When it reaches (5,2), the robot turns 90° counterclockwise and continues to move linearly. After the turn, what is the y -coordinate of the robot's position when its x-coordinate is 4?
17. A robot begins moving on a linear path at (3, 1). When it reaches (5,2), the robot turns 90° counterclockwise and continues to move linearly. After the turn, what is the y -coordinate of the robot's position when its x-coordinate is 4?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 17RE
Related questions
Question
![### Problem Statement
17. A robot begins moving on a linear path at (3, 1). When it reaches (5, 2), the robot turns 90° counterclockwise and continues to move linearly. After the turn, what is the y-coordinate of the robot's position when its x-coordinate is 4?
---
### Analysis
Let's break this down step by step:
1. **Initial Movement**:
- The robot moves from point \( (3, 1) \) to point \( (5, 2) \).
2. **Direction of Initial Path**:
- Calculate the slope of the path from \( (3, 1) \) to \( (5, 2) \):
\[
\text{slope} = \frac{2 - 1}{5 - 3} = \frac{1}{2}
\]
- The direction of this path is \( \frac{1}{2} \).
3. **90° Counterclockwise Turn**:
- A 90° counterclockwise turn means the direction perpendicular to the original direction. The slope of the new path will be the negative reciprocal of the original slope:
\[
\text{new slope} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]
4. **Equation of New Path**:
- We use the point-slope form of the line equation to find the equation of the new path through point \( (5, 2) \):
\[
y - 2 = -2(x - 5)
\]
Simplify the equation:
\[
y - 2 = -2x + 10
\]
\[
y = -2x + 12
\]
5. **Finding the y-coordinate when x = 4**:
- Substitute \( x = 4 \) into the equation of the new path:
\[
y = -2(4) + 12
\]
\[
y = -8 + 12
\]
\[
y = 4
\]
### Conclusion
The y-coordinate of the robot's position when its x-coordinate is 4 is \(\boxed{4}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa05adf2c-7d26-4e8f-9974-7b0b186f54f9%2F45fe3c24-91f6-49e3-9784-a05aff701f45%2Fdzpyo7a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
17. A robot begins moving on a linear path at (3, 1). When it reaches (5, 2), the robot turns 90° counterclockwise and continues to move linearly. After the turn, what is the y-coordinate of the robot's position when its x-coordinate is 4?
---
### Analysis
Let's break this down step by step:
1. **Initial Movement**:
- The robot moves from point \( (3, 1) \) to point \( (5, 2) \).
2. **Direction of Initial Path**:
- Calculate the slope of the path from \( (3, 1) \) to \( (5, 2) \):
\[
\text{slope} = \frac{2 - 1}{5 - 3} = \frac{1}{2}
\]
- The direction of this path is \( \frac{1}{2} \).
3. **90° Counterclockwise Turn**:
- A 90° counterclockwise turn means the direction perpendicular to the original direction. The slope of the new path will be the negative reciprocal of the original slope:
\[
\text{new slope} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]
4. **Equation of New Path**:
- We use the point-slope form of the line equation to find the equation of the new path through point \( (5, 2) \):
\[
y - 2 = -2(x - 5)
\]
Simplify the equation:
\[
y - 2 = -2x + 10
\]
\[
y = -2x + 12
\]
5. **Finding the y-coordinate when x = 4**:
- Substitute \( x = 4 \) into the equation of the new path:
\[
y = -2(4) + 12
\]
\[
y = -8 + 12
\]
\[
y = 4
\]
### Conclusion
The y-coordinate of the robot's position when its x-coordinate is 4 is \(\boxed{4}\).
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