Trigonometry (MindTap Course List)
10th Edition
ISBN: 9781337278461
Author: Ron Larson
Publisher: Cengage Learning
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![Consider an object moving in the plane whose location at time t seconds is given by the parametric equations:
x(t)=5cos(at)
y(t)=3sin(t).
Assume the distance units in the plane are meters.
(a) The object is moving around an ellipse with equation:
+2
+=1
where a= 5
and b= 3
(b) The location of the object at time t=1/3 seconds is
(2.5
2.598
(c) The horizontal velocity of the object at time t is x ' (t)=
—5nsin(t)
m/s.
(d) The horizontal velocity of the object at time t=1/3 seconds is
(e) The vertical velocity of the object at time t is y' (t)= 3π cos(л)
m/s.
m/s.
(f) The vertical velocity of the object at time t=1/3 seconds is 4.712
(g) The slope of the tangent line at time t=1/3 seconds is -0.347
m/s.
(h) Recall, the speed of the object at time t is given by the equation:
s(t)=√√ [x '(t)]² + [y ' (t)]² m/s.
The speed of the object at time t=1/3 seconds is
(i) The first time when the horizontal and vertical velocities are equal is time t= 0.828
(j) Let Q be the position of the object at the time you found in part (i). The slope of the tangent line to the ellipse at Q is](https://content.bartleby.com/qna-images/question/813110af-3609-46c1-b426-a57c20af71d4/e9d3ee44-acd3-4013-8704-7746301c6676/21b0rn_processed.png)
Transcribed Image Text:Consider an object moving in the plane whose location at time t seconds is given by the parametric equations:
x(t)=5cos(at)
y(t)=3sin(t).
Assume the distance units in the plane are meters.
(a) The object is moving around an ellipse with equation:
+2
+=1
where a= 5
and b= 3
(b) The location of the object at time t=1/3 seconds is
(2.5
2.598
(c) The horizontal velocity of the object at time t is x ' (t)=
—5nsin(t)
m/s.
(d) The horizontal velocity of the object at time t=1/3 seconds is
(e) The vertical velocity of the object at time t is y' (t)= 3π cos(л)
m/s.
m/s.
(f) The vertical velocity of the object at time t=1/3 seconds is 4.712
(g) The slope of the tangent line at time t=1/3 seconds is -0.347
m/s.
(h) Recall, the speed of the object at time t is given by the equation:
s(t)=√√ [x '(t)]² + [y ' (t)]² m/s.
The speed of the object at time t=1/3 seconds is
(i) The first time when the horizontal and vertical velocities are equal is time t= 0.828
(j) Let Q be the position of the object at the time you found in part (i). The slope of the tangent line to the ellipse at Q is
Transcribed Image Text:This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one.
A 10-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 5 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The wheel rotates counterclockwise at 1.5
rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture.
B
A
B
A
B
at some instant, the piston will be tangent to the circle
(a) Express the x and y coordinates of point A as functions of t:
x= 5 cos(3)
and y= 5 sin(3πt)
(b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds:
-cot (3πt)
(c) Express the x-coordinate of the right end of the rod at point B as a function of t: 5 cos(3лt) + 10 sin(3лt)
(d) Express the slope of the rod as a function of t: -cot (3πt)
(c) Find the first time when the rod is tangent to the circle:
seconds.
(d) At the time in (c), what is the slope of the rod?
(e) Find the second time when the rod is tangent to the circle:
seconds.
(f) At the time in (e), what is the slope of the rod?
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