Question

A wheel with radius 5 cm is being pushed up a ramp at a rate of 8 cm per second. The ramp is 500 cm long, and 280 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).

Cross-sectional diagram of a ramp with a wheel at the base. The shape is a right triangle, and the hypotenuse and vertical sides are labeled with the given length and height. Point P is at the topmost point of the wheel.
500 cm280 cm


Write parametric equations for the position of the point P as a function of t, time in seconds after the ball starts rolling up the ramp. Both
x
and
y
are measured in centimeters.


x
=



y
=

Answer 1

when the wheel has rotated through an angle θ, then if the center of the circle is at (h,k) we have
x = h+5cosθ
y = k-5sinθ
so the real trick is, what are θ,h,k in terms of time t?
dθ = 10πcm/8s = 5π/4 rad/s
so θ = 5π/4 t (it starts with θ(0) = 0)
The ramp makes an angle Ø such that tanØ = 280/500
When the wheel starts to roll up the plank, the radius perpendicular to the plank has location (h,k) where
h = -5sinØ
k = 5cosØ
and the horizontal and vertical speeds are
8cosØ and 8sinØ
when you put that all together, you might consider comparing it to the equations of a cycloid:
x = r(θ-sinθ)
y = r(1-cosθ)
adjusted for the sloping ramp.

If you get stuck, come on back with what you got.

Answer 2

my answer got wrong and I got new one still couldn't figure out the right answer

A wheel with a radius 2 cm is being pushed up a ramp at a rate of 7 cm per second. The ramp is 700 cm long, and 140 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).

Cross-sectional diagram of a ramp with a wheel at the base. The shape is a right triangle, and the hypotenuse and vertical sides are labeled with the given length and height. Point P is at the topmost point of the wheel.
700 cm140 cm


Write parametric equations for the position of the point P as a function of t, time in seconds after the ball starts rolling up the ramp. Both
x
and
y
are measured in centimeters.


x
=



y
=



You will have a radical expression for part of the horizontal component. It's best to use the exact radical expression even though the answer that WAMAP shows will have a decimal approximation.