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. A B 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(3nt) 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 (3πt) (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?

Find parametric equations for the trajectory of a particle that moves along the circumference:

A bee fles around a room at an altitude of 2= 2.3 meters making circles, so that its r coordinate varies from 1.3 to 1.9 meters, and its y coordinate varies from 2.2 to 2.8 meters. The bee completes one circle every 2.3 seconds. At sime=0 seconds, the bee's y coordinate is 2.8 meters and its velocity vector is completely in the +i direction. Write parametric equations to describe the motion of the bee z(t)= sin(2p/23)+22 y(t) =30(2pit2:3)+1.9 z(t) = 23

Write the parametric equations a = 4t – t, y = 2 – 3t in the given Cartesian form.

A shot put being thrown is represented by the parametric equations: x = 22.98t y = -16t² + 19.3t+6 If a 6 foot man stands 25 feet away from thrower, will the shot put hit him? If not, by how much would the shot put miss the 6 foot man? Round to the nearest hundredths place. If not applicable, then write NA. feet

A major league baseball pitcher throws a pitch that follows these parametric equations: x(t) = 146t y(t)=-16t² + 6 + 5. The time units are seconds and the distance units are feet. The distance between the location of the pitcher and homeplate (where the batter stands) is 60.5 feet. Give EXACT answers, unless instructed otherwise. (a) Calculate the horizontal velocity of the baseball at time t; this is the function x'(t)= | 146 (b) What is the horizontal velocity of the baseball when it passes over homeplate? 146 (c) What is the vertical velocity of the baseball at time t; this is the function y'(t)=-32t+6 (d) Recall that the speed of the baseball at time t is s(t)=√ [x '(t)]² + [y' (t)]² ft/sec. What is the speed of the baseball (in mph) when it passes over homeplate? (e) At what time does the baseball hit the ground, assuming the batter and catcher miss the ball? (f) What is the magnitude of the angle at which the baseball hits the ground? mph. ft/sec. ft/sec ft/sec. sec. rad. (This…

the plane curve described by the parametric equations x=cost and y=3sint has a counter-clockwise rotation. Alter one or both equations so that you obtain the same curve with the opposite orientation

Part D, H, J