**Understanding the Safety in Vehicle Race Tracks: A Calculus Application****Graphs and Diagrams Analysis:**- **Figure 3.21** - **Part (a)**: This graph depicts the function $ f(x) = x^3 + 3x^2 + x $. - The graph exhibits a cubic curve with several turning points, located on a standard Cartesian coordinate plane with x-axis ranging from -5 to 5 and y-axis ranging from -5 to 5. - **Part (b)**: This graph represents the same cubic function, $ f(x) = x^3 + 3x^2 + x $, but includes additional elements. - A visual representation of the grandstand is illustrated at the coordinates (-1.9, 2.8). - This diagram particularly highlights the section of the racetrack near the point where the turning point of the function—the slope of the tangent line—is 1.**Educational Problems and Questions:**1. **Problem Statement**: - Physicists have observed that drivers are most likely to lose control of their vehicles when navigating a turn, specifically at the juncture where the slope of the tangent line to the curve is 1. First, determine the $(x, y)$ coordinates of this key point on the racetrack.2. **Equation of Tangent Line**: - Derive the equation for the tangent line to the curve at the identified point.3. **Spectator Safety Analysis**: - Assess whether the spectators, located at $(x,y) = (-1.9, 2.8)$, are in a potential danger zone. Determine this by finding the x-coordinate where the tangent line intersects $y = 2.8$ and decide if this point is safely positioned relative to the grandstand.4. **Additional Safety Scenario**: - Consider an alternative scenario where a driver loses control earlier than anticipated, at the coordinates $(-2.5, 0.625)$. Calculate the slope of the tangent line at this point.5. **Safety Confirmation**: - Based on the tangent line slope derived in part 4, analyze whether the spectators remain safe in this scenario.6. **Final Decision Making**: - Conclude if the current grandstand placement is adequate for ensuring spectator safety, or if a redesign or relocation is warranted based on the mathematical insights gained

Answered: Image /qna-images/answer/e1d95779-3d16-4ba3-9766-5c5a4c6d9422.jpg

1. Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the (x, y) coordinates of this point near the turn. 2. Find the equation of the tangent line to the curve at this point. 3. To determine whether the spectators are in danger in this scenario, find the x-coordinate of the point where the tangent line crosses the line y = 2.8. Is this point safely to the right of the grandstand? Or are the spectators in danger? %3D

1. Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the (x, y) coordinates of this point near the turn. 2. Find the equation of the tangent line to the curve at this point. 3. To determine whether the spectators are in danger in this scenario, find the x-coordinate of the point where the tangent line crosses the line y = 2.8. Is this point safely to the right of the grandstand? Or are the spectators in danger? %3D

Calculus: Early Transcendentals

8th Edition

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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Question

**Understanding the Safety in Vehicle Race Tracks: A Calculus Application**

**Graphs and Diagrams Analysis:**

- **Figure 3.21**
- **Part (a)**: This graph depicts the function $ f(x) = x^3 + 3x^2 + x $.
- The graph exhibits a cubic curve with several turning points, located on a standard Cartesian coordinate plane with x-axis ranging from -5 to 5 and y-axis ranging from -5 to 5.
- **Part (b)**: This graph represents the same cubic function, $ f(x) = x^3 + 3x^2 + x $, but includes additional elements.
- A visual representation of the grandstand is illustrated at the coordinates (-1.9, 2.8).
- This diagram particularly highlights the section of the racetrack near the point where the turning point of the function—the slope of the tangent line—is 1.

**Educational Problems and Questions:**

1. **Problem Statement**:
- Physicists have observed that drivers are most likely to lose control of their vehicles when navigating a turn, specifically at the juncture where the slope of the tangent line to the curve is 1. First, determine the $(x, y)$ coordinates of this key point on the racetrack.

2. **Equation of Tangent Line**:
- Derive the equation for the tangent line to the curve at the identified point.

3. **Spectator Safety Analysis**:
- Assess whether the spectators, located at $(x,y) = (-1.9, 2.8)$, are in a potential danger zone. Determine this by finding the x-coordinate where the tangent line intersects $y = 2.8$ and decide if this point is safely positioned relative to the grandstand.

4. **Additional Safety Scenario**:
- Consider an alternative scenario where a driver loses control earlier than anticipated, at the coordinates $(-2.5, 0.625)$. Calculate the slope of the tangent line at this point.

5. **Safety Confirmation**:
- Based on the tangent line slope derived in part 4, analyze whether the spectators remain safe in this scenario.

6. **Final Decision Making**:
- Conclude if the current grandstand placement is adequate for ensuring spectator safety, or if a redesign or relocation is warranted based on the mathematical insights gained

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