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 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?

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 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?

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.4: Hyperbolas

Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.

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