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?

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

Find a set of parametric equations for the rectangular equation y=4x-4 that satisfies the condition at the point (4, 12). t=0 Select one: a. x=t, y = 4t-4 b. x=t, y=31²+1-4 c. x=t+4, y = 31² + 5t- 8 O d. x=-4, y = 41+ 12 e. xt+4, y = 4t+ 12 Check

Find a parametric equation from the given conditions: Parabola with vertex at (2,-3) and opens downward.

Part D, H, J

Please do the following handwritten

Write the Cartesian equation of the line given by the parametric equations (t) = y(t) 5 – 4t || -1- 6t . z(t) 6t Enter the terms of the equation in alphabetic order (that is, first the term with x, followed by the term with y and then the term with z. =

Convert the line given by the parametric equations y(t) Enter the symmetric equations in alphabetic order. (x(t) = -4+6t = 3-t (z(t) = 5-7t to symmetric equations.

Given the following parametric equations, convert to rectangular equations: x : = t+ 5 and y = t² – 11. Write the equation in standard form. Use the ^ symbol for adding exponents. Example: Type x^2 for x2. No spaces! type your answer...

Find the parametric equations of the line r = ⟨1, 2, 3⟩ + t(4, 0, -6⟩.