Eliminate the parameter and describe the shape of the curve: :=√√x+4; _y=3√t; for 0≤t≤16
Eliminate the parameter and describe the shape of the curve: :=√√x+4; _y=3√t; for 0≤t≤16
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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![**Eliminate the parameter and describe the shape of the curve:**
Given the parametric equations:
\[ x = \sqrt{t} + 4, \quad y = 3\sqrt{t} \]
for \[ 0 \leq t \leq 16 \].
To eliminate the parameter \( t \), we solve for \( t \) in terms of \( x \) or \( y \).
1. From the equation \( x = \sqrt{t} + 4 \):
\[
\sqrt{t} = x - 4
\]
\[
t = (x - 4)^2
\]
2. Substitute \( t \) in the equation for \( y \):
\[
y = 3 \sqrt{t} = 3(x - 4)
\]
Therefore, the Cartesian equation of the curve is:
\[
y = 3(x - 4)
\]
The shape of this curve is a straight line with a slope of 3, offset horizontally by 4 units. The line is defined for the range \( 0 \leq t \leq 16 \), which translates to \( 0 \leq x-4 \leq 4 \), or \( 4 \leq x \leq 8 \).
The corresponding range for \( y \) is \( 0 \leq y \leq 12 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F135565b3-f69f-45e7-bb45-4192b3629324%2Fe2d48ac0-2a6d-470d-b9db-c34dd5b8b147%2Fjne3ptr_processed.png&w=3840&q=75)
Transcribed Image Text:**Eliminate the parameter and describe the shape of the curve:**
Given the parametric equations:
\[ x = \sqrt{t} + 4, \quad y = 3\sqrt{t} \]
for \[ 0 \leq t \leq 16 \].
To eliminate the parameter \( t \), we solve for \( t \) in terms of \( x \) or \( y \).
1. From the equation \( x = \sqrt{t} + 4 \):
\[
\sqrt{t} = x - 4
\]
\[
t = (x - 4)^2
\]
2. Substitute \( t \) in the equation for \( y \):
\[
y = 3 \sqrt{t} = 3(x - 4)
\]
Therefore, the Cartesian equation of the curve is:
\[
y = 3(x - 4)
\]
The shape of this curve is a straight line with a slope of 3, offset horizontally by 4 units. The line is defined for the range \( 0 \leq t \leq 16 \), which translates to \( 0 \leq x-4 \leq 4 \), or \( 4 \leq x \leq 8 \).
The corresponding range for \( y \) is \( 0 \leq y \leq 12 \).
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