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.
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
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
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?
To find the values of 't' during the first 30 seconds when the pedal is more than 30.5 cm above the ground, we can use the equation you've mentioned: y = 19sin[(π/10)(t-5)] + 40.
First, let's set up the inequality that represents the condition: y > 30.5.
19sin[(π/10)(t-5)] + 40 > 30.5
Next, let's isolate the sine term by subtracting 40 from both sides:
19sin[(π/10)(t-5)] > 30.5 - 40
19sin[(π/10)(t-5)] > -9.5
Now, divide both sides by 19:
sin[(π/10)(t-5)] > -9.5/19
simplify:
sin[(π/10)(t-5)] > -0.5
To find the values of 't' that satisfy this inequality, let's find the solutions of the equation sin[(π/10)(t-5)] = -0.5.
sin[(π/10)(t-5)] = -0.5
To find the solutions, we need to find the angles whose sine is -0.5. This occurs at two standard angles: -π/6 and 7π/6.
(π/10)(t-5) = -π/6 or (π/10)(t-5) = 7π/6
Solve each equation for 't':
(π/10)(t-5) = -π/6
t - 5 = -6π/10
t = -6π/10 + 5
(π/10)(t-5) = 7π/6
t - 5 = (7π/6)(10/π)
t = 7(5/3)
t = 35/3
Now, let's check if these values of 't' fall within the first 30 seconds (0 to 30):
-6π/10 + 5 is approximately 2.8, which is not within the range of 0 to 30.
35/3 is approximately 11.7, which is within the range of 0 to 30.
Therefore, the solution is t = 35/3, which means during the first 30 seconds of riding, the pedal is more than 30.5 cm above the ground at approximately 11.7 seconds.