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 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
To find out when the pedal is more than 30.5 cm above the ground during the first 30 seconds of riding, we need to determine the time intervals during which the pedal's height is above 30.5 cm.
First, let's calculate the range of pedal height values over the 60-second cycle. The minimum height recorded is 21 cm, and the maximum height is 59 cm, so the range is 59 - 21 = 38 cm.
Next, we need to determine the height of the pedal at each time interval of the 60-second cycle. Since there are 3 cycles in 60 seconds, each cycle takes 60/3 = 20 seconds.
Dividing the range of pedal height (38 cm) by the time it takes for a full cycle (20 seconds), we can find the rate of change for the pedal's height: 38 cm / 20 s = 1.9 cm/s.
Now, we can determine the time intervals during which the pedal's height is more than 30.5 cm.
During the first 20 seconds, the pedal will rise 1.9 cm/s x 20 s = 38 cm, which is above 30.5 cm. So, the pedal is more than 30.5 cm above the ground for the entire first 20-second interval.
During the next 20 seconds (from 20 to 40 seconds), the pedal will rise an additional 38 cm. So, the pedal is still above 30.5 cm for the entire second 20-second interval as well.
Finally, during the last 10 seconds (from 40 to 50 seconds), the pedal will rise half of the rate of change in the previous time intervals: 1.9 cm/s x 10 s = 19 cm. Since this is less than 30.5 cm, the pedal is not above 30.5 cm during the last 10 seconds of the first 30 seconds of riding.
Therefore, the pedal is more than 30.5 cm above the ground for the entire first 20 seconds and the entire second 20 seconds, which means the pedal is above 30.5 cm for the first 40 seconds of riding.
To summarize, during the first 30 seconds of riding, the pedal is more than 30.5 cm above the ground for the first 20 seconds.