PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
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Expert Solution & Answer
Chapter 8.2, Problem 73E
Solution
(a.)
The plot of (x,y) relation for velocity versus position for 0≤t≤2π .
The plot of (x,y) relation for velocity versus position for 0≤t≤2π has been provided.
Given:
As a pendulum swings toward and away from a motion detector, its distance (in meters) from the detector is given by the position function x(t)=3+cos(2t−5) , where t represents time (in seconds). The velocity (in m/s) of the pendulum is given by y(t)=−2sin(2t−5) .
Concept used:
Parametric equations x(t) and y(t) can be plotted on the xy plane for a given bound of t .
Calculation:
The given parametric equations are x(t)=3+cos(2t−5) and y(t)=−2sin(2t−5) .
Plotting these parametric equations on the xy
plane as follows:
Conclusion:
The plot of (x,y) relation for velocity versus position for 0≤t≤2π has been provided.
(b.)
The equation of the resulting conic in standard form, in terms of x and y , and eliminating the parameter t .
It has been determined that the equation of the resulting conic in standard form, in terms of x and y , and eliminating the parameter t , is (x−3)212+y222=1 .
Given:
As a pendulum swings toward and away from a motion detector, its distance (in meters) from the detector is given by the position function x(t)=3+cos(2t−5) , where t represents time (in seconds). The velocity (in m/s) of the pendulum is given by y(t)=−2sin(2t−5) .
Concept used:
Parametric equations x(t) and y(t) can be plotted on the xy plane for a given bound of t .
Calculation:
The given parametric equations are x(t)=3+cos(2t−5) and y(t)=−2sin(2t−5) .
Simplifying,
x−3=cos(2t−5) ,
y2=−sin(2t−5)
Squaring both sides of each equation,
(x−3)2=cos2(2t−5) ,
y24=sin2(2t−5)
Adding both equations,
(x−3)2+y24=cos2(2t−5)+sin2(2t−5)
Using trigonometric identity; sin2θ+cos2θ=1 ,
(x−3)2+y24=1
Converting to standard form,
(x−3)212+y222=1
This is the required equation of the resulting conic in standard form.
Conclusion:
It has been determined that the equation of the resulting conic in standard form, in terms of x and y , and eliminating the parameter t , is (x−3)212+y222=1 .
Chapter 8 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
Ch. 8.1 - Prob. 1QRCh. 8.1 - Prob. 2QRCh. 8.1 - Prob. 3QRCh. 8.1 - Prob. 4QRCh. 8.1 - Prob. 5QRCh. 8.1 - Prob. 6QRCh. 8.1 - Prob. 7QRCh. 8.1 - Prob. 8QRCh. 8.1 - Prob. 9QRCh. 8.1 - Prob. 10QR
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