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

I was able to get an equation, y=19sin[pi/10(t-5)]+40 and solve for t=10/3, but how do I find the other t values

Answer 1

radius = (59-21)/2 = 19 agree
center = 21 + 19 = 40 agree
T = period = 60/3 = 20 seconds
so
y = 40 + 19 sin (2 pi t/T - phase) = 40 + 19 sin (pi t/10 - phase)
whet t = 0 ,,, sin (pi t/10 - phase) = -1
so sin (- phase) = -1
- phase = -pi/2
so
y = 40 + 19 sin (pi t/10 - pi/2) agreed
we need y > 30.5
19 sin (pi t/10 - pi/2) > - 9.5
sin (pi t/10 - pi/2) > - 0.5
that is from the angle = -30 deg to the angle 180+30 = 210 deg
or
- pi/6 to (pi + pi/6) = -pi/6 to 7 pi/6

pi t/10 - pi/2 = -pi/6
t/10 - 1/2 = - 1/6
t/10 = 1/3
t = 10/3 agree
then
pi t/10 - pi/2 = 7 pi/6
t/10 = 3/6 + 7/6 = 10/6
t = 100/6 = 16.7 seconds

Answer 2

I did this for you yesterday, and did you notice that Damon got the same answer?
https://www.jiskha.com/questions/1829545/the-height-of-the-pedals-of-a-bicycle-change-with-respect-to-time-the-minimum-height

You also claimed that "I was able to get an equation, y=19sin[pi/10(t-5)]+40"
That's not nice!
Why did you change names from Madison?