Question

Dave is going for a ride on his unicycle. The radius of the wheel is 25.5cm. When he gets on, the valve of the wheel is at its maximum height. He pedals along a path at a speed of 30km/h.

At what height will the valve be located after he has cycled for 4 minutes?

How do I find the period of this? I think try finding the equation first. Thanks!

Answer 1

One rotation of the wheel would cover
2(25.5)π or 51π cm

30 km/hr = 30(1000)(100) cm / 60 minutes
= 50000 cm/min

in 4 min Dave will have covered 4(50000) cm
= 200,000 cm (using Distance = rate x time )
So the number of rotations = 200000/(51π)
= 1248.274063

The valve will be .274063th of a rotation or 98.663° from the vertical
or 8.663° down from the vertical
sin 8.663 = x/25.5
x = 3.459
so the valve is at 25.5-3.459 cm
or at a height of 22.04 cm

or ....
amplitude is 25.5

for period:
need time for one rotation
length of one rotation = 51π (see above)
speed = 50,000 cm/min
time for one rotation = dist/speed
= 51π/50000

we know the equation must be something like
Height = 25.5cos kt + 25.5

where 2π/k =51π/50000
k = 100,000/51

height = 25.5 cos ((100,000/51)t) + 25.5
set calculator to radians, sub t = 4
to get
height = 21.659

argghh, small discrepancy in answer.
(can't seem to find my error)