The height of the pedals of a bicycle change with respect to time. The minimum height recorded for the pedals was 21 cm above the ground, and the maximum height was 59 cm. Assume the bicycle is peddled at the rate of 3 cycles in 60 seconds, and that the pedal starts at its lowest possible position. When during the first 30 seconds of riding is the pedal more than 30.5 cm above the ground? Round answers to 1 decimal place.

Question

Reiny · Accepted Answer

I will assume that we are looking at the position of only one of the pedals .
Each of the 2 pedals follows a sinusoidal path, (sine or cosine)

let's do a sine curve.
the amplitude is (59-21)/2 = 19 cm
so let's start with height = 19sin(?)
3 cylces takes 60 seconds, so 1 cycle = 20 sec
2π/k = 20
k = π/10
then height = 19sin(π/10t)
we have to raise this by 40 to get a min of 21 and a max of 59
height = 19sin(π/10 t) + 40
sofar I got:
https://www.wolframalpha.com/input/?i=graph+y+%3D+19sin%28%CF%80%2F10+t%29+%2B+40
Notice our min occurs when t = -5, we want our min to happen when t = 0, so we have to move
our curve to the right 5 units
height = 19sin (π/10(t - 5)) + 40

https://www.wolframalpha.com/input/?i=graph+y+%3D+19sin+%28%CF%80%2F10%28t+-+5%29%29+%2B+40+

That's better.

So now we want 19sin (π/10(t - 5)) + 40 = 30.5
sin(π/10(t - 5)) = -.5
I know sin(-π/6) = -.5
π/10(t-5) = -π/6
1/10(t-5) = -1/6
times 30
3(t-5) = -5
3t - 15 = -5
t = 10/3 , looks good on my graph
because of the symmetry or basic trig, the other value in the first cycle is 20-10/3 = 50/3 seconds

Your question deals with the first 30 seconds, which is 1 1/2 cycles.
So a pedal is above 30.5 cm between 10/3 and 50/3 seconds, that is for 40/3 seconds, and then
another 10-10/3 or 20/3 seconds in the next half cycle