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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![### Problem Statement: Intersection of a Parametric Line with the \(xz\)-Plane
Given the parametric equations of a line:
\[
\begin{cases}
x(t) = -6 - 7t \\
y(t) = 7 + t \\
z(t) = -5 - 4t
\end{cases}
\]
we are tasked with finding the point at which this line intersects the \(xz\)-plane.
### Explanation
To determine where the line intersects the \(xz\)-plane, we need to find the value of the parameter \(t\) at which the \(y\)-component of the line is zero (i.e., \(y(t) = 0\)).
1. Set the \(y\)-component equation to zero:
\[
7 + t = 0
\]
2. Solve for \(t\):
\[
t = -7
\]
3. Substitute \(t = -7\) into the equations for \(x(t)\) and \(z(t)\) to find the corresponding \(x\) and \(z\) coordinates:
\[
x(-7) = -6 - 7(-7) = -6 + 49 = 43
\]
\[
z(-7) = -5 - 4(-7) = -5 + 28 = 23
\]
Therefore, the point at which the line intersects the \(xz\)-plane is \((43, 0, 23)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d3466e-6a31-4aff-972b-e33941523058%2Fad92129d-b265-4d8f-adb8-3f3df89096e9%2Fj2v8ov_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Intersection of a Parametric Line with the \(xz\)-Plane
Given the parametric equations of a line:
\[
\begin{cases}
x(t) = -6 - 7t \\
y(t) = 7 + t \\
z(t) = -5 - 4t
\end{cases}
\]
we are tasked with finding the point at which this line intersects the \(xz\)-plane.
### Explanation
To determine where the line intersects the \(xz\)-plane, we need to find the value of the parameter \(t\) at which the \(y\)-component of the line is zero (i.e., \(y(t) = 0\)).
1. Set the \(y\)-component equation to zero:
\[
7 + t = 0
\]
2. Solve for \(t\):
\[
t = -7
\]
3. Substitute \(t = -7\) into the equations for \(x(t)\) and \(z(t)\) to find the corresponding \(x\) and \(z\) coordinates:
\[
x(-7) = -6 - 7(-7) = -6 + 49 = 43
\]
\[
z(-7) = -5 - 4(-7) = -5 + 28 = 23
\]
Therefore, the point at which the line intersects the \(xz\)-plane is \((43, 0, 23)\).
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