How do you find parametric equations for the path of a particle that moves along the #x^2 + (y-3)^2 = 16# once around clockwise, starting at (4,3) 0 ≤ t ≤ 2pi ?

Answer 1

#x = 4 cos t#
#y = 3 + 4 sin t#

#t in [0, 2 pi]#

this is a circle of radius 4 centred on (0,3)

we can use polar coordinates but with a twist to reflect the fact that the circle is not centred about the origin

thus we have

#x = 4 cos t# #y = 3 + 4 sin t#
#implies sin^2 t + cos^2 t = (x/4)^2 + ((y-3)/4)^2 = 1#!!
this satisfies the starting condition as #((x_o),(y_o)) = ((4 cos 0),(3 + 4 sin 0))= ((4),(3))#

it will move clockwise around the circle centre. in polar, angle t is measured clockwise from the x-axis.

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Answer 2

Parametric equations for the path of a particle moving along the given curve can be found using the parametric equations for a circle. The parametric equations for a circle centered at (h, k) with radius r are:

x = h + r * cos(t) y = k + r * sin(t)

For the given curve x^2 + (y - 3)^2 = 16, the center is (h, k) = (0, 3) and the radius is 4. Thus, the parametric equations become:

x = 4 * cos(t) y = 3 + 4 * sin(t)

To start the particle at (4, 3), adjust the initial value of t accordingly:

When t = 0, the particle is at (4, 3), so we use t = 0 as the starting point. When t = 2π, the particle has completed one clockwise revolution around the circle.

Therefore, the parametric equations for the path of the particle are:

x(t) = 4 * cos(t) y(t) = 3 + 4 * sin(t) for 0 ≤ t ≤ 2π.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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