ther the space curve given by r(t) = (2t², t² + 2t - 8, t - 2) intersects the x-axis, and if it do se symbolic notation and fractions where needed. Give your answer as the coordinates of a point in the form O SOLUTION if the curve does not intersect the x-axis.) mere. point coordinates:
ther the space curve given by r(t) = (2t², t² + 2t - 8, t - 2) intersects the x-axis, and if it do se symbolic notation and fractions where needed. Give your answer as the coordinates of a point in the form O SOLUTION if the curve does not intersect the x-axis.) mere. point coordinates:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Educational Task: Intersection of a Space Curve with the x-axis**
Determine whether the space curve given by \( \mathbf{r}(t) = \langle 2t^2, \ t^2 + 2t - 8, \ t - 2 \rangle \) intersects the x-axis, and if it does, determine where.
(Use symbolic notation and fractions where needed. Give your answer as the coordinates of a point in the form \((*, *, *)\). Enter NO SOLUTION if the curve does not intersect the x-axis.)
**Input box:**
Point coordinates: ________________
---
**Explanation:**
The problem involves finding the intersection points of the given space curve \( \mathbf{r}(t) \) with the x-axis. For a point to lie on the x-axis, both the y and z components of the vector must be zero. Therefore, solve for \( t \) where:
\[ t^2 + 2t - 8 = 0 \]
\[ t - 2 = 0 \]
If any solutions exist, substitute the value of \( t \) found into the x-component to find the coordinates. If no solution is found, enter "NO SOLUTION" in the input box provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd370dc10-a80b-4ff7-aff7-eb45807dd9a9%2F9d5aee99-5545-420f-990e-e07bbec65d3a%2Fo40ee3b_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Task: Intersection of a Space Curve with the x-axis**
Determine whether the space curve given by \( \mathbf{r}(t) = \langle 2t^2, \ t^2 + 2t - 8, \ t - 2 \rangle \) intersects the x-axis, and if it does, determine where.
(Use symbolic notation and fractions where needed. Give your answer as the coordinates of a point in the form \((*, *, *)\). Enter NO SOLUTION if the curve does not intersect the x-axis.)
**Input box:**
Point coordinates: ________________
---
**Explanation:**
The problem involves finding the intersection points of the given space curve \( \mathbf{r}(t) \) with the x-axis. For a point to lie on the x-axis, both the y and z components of the vector must be zero. Therefore, solve for \( t \) where:
\[ t^2 + 2t - 8 = 0 \]
\[ t - 2 = 0 \]
If any solutions exist, substitute the value of \( t \) found into the x-component to find the coordinates. If no solution is found, enter "NO SOLUTION" in the input box provided.
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