The height of a person above the ground on a Ferris wheel is given by the function h(t)=9 sin {pi/20 (t- pi/2))+ 12 where h(t) is the height in meters of person t seconds after getting on
a) how long does it take the Ferris wheel to make one complete revolution
b)what is the height of the ferris wheel if the riders get on at 15s
the period of sin(kt) is 2π/k
So, since your k here is π/20, the period is (2π)/(π/20) = 40 seconds.
the height of the wheel does not depend on when the riders embark...
However, the height at the top is the center-line (12) + the amplitude (9)
Ahh. I think you must mean the height of the car when the riders embark at t=15. Well, just substitute t=15 into the function and evaluate h(15).
a) To find how long it takes for the Ferris wheel to make one complete revolution, we need to determine the period of the function h(t).
The general form of a sinusoidal function is h(t) = A sin(B(t - C)) + D, where A, B, C, and D are constants.
Comparing this form with the given equation h(t) = 9 sin((π/20)(t - π/2)) + 12, we can see that the coefficient of (t - C) is (π/20) which corresponds to B.
The period of a sinusoidal function is given by the formula period = (2π) / |B|.
In our equation, |B| = |π/20| = π/20.
Therefore, the period of the function h(t) is period = (2π) / (π/20) = (2π) * (20/π) = 40 seconds.
Hence, it takes the Ferris wheel 40 seconds to make one complete revolution.
b) To find the height of the Ferris wheel when the riders get on at 15 seconds, we need to substitute t = 15 into the equation h(t) = 9 sin((π/20)(t - π/2)) + 12.
h(15) = 9 sin((π/20)(15 - π/2)) + 12
= 9 sin((π/20)(15 - (π/2))) + 12
= 9 sin((π/20)(15 - (π/2))) + 12
= 9 sin((π/20)(15 - (π/2))) + 12
= 9 sin((π/20)(15 - 10π/20)) + 12
= 9 sin((π/20)(15 - 10π/20)) + 12
= 9 sin((π/20)(15 - 10π/20)) + 12
= 9 sin(3π/4) + 12
= 9 * √2/2 + 12
= 9√2/2 + 12
= 4.5√2 + 12.
Therefore, the height of the Ferris wheel when the riders get on at 15 seconds is 4.5√2 + 12 meters.
To solve this problem, we need to analyze the given function and use the properties of the sine function.
a) How long does it take the Ferris wheel to make one complete revolution?
We know that one complete revolution of a Ferris wheel corresponds to one complete period of the sine function. The period of a sine function can be calculated using the formula:
Period = (2 * π) / |b|
In our case, b = π/20. Substituting this value into the above formula, we get:
Period = (2 * π) / |π/20|
To simplify, we divide π by π/20, which is equivalent to multiplying by 20/π:
Period = (2 * π) * (20/π)
The π cancels out, leaving us with:
Period = 40
Therefore, it takes 40 seconds for the Ferris wheel to make one complete revolution.
b) What is the height of the Ferris wheel if the riders get on at 15 seconds?
To find the height of the Ferris wheel at a specific time, we need to substitute the given time into the height function:
h(t) = 9sin(π/20 (t - π/2)) + 12
Substituting t = 15 into the function, we get:
h(15) = 9sin(π/20 (15 - π/2)) + 12
We can simplify the inside of the sine function:
15 - π/2 = 30/2 - π/2 = 15 - (π/2) = 15π/2
Substituting this into the function:
h(15) = 9sin(π/20 * (15π/2)) + 12
Further simplification using basic trigonometric identities:
h(15) = 9sin(3π/2) + 12
We know that sin(3π/2) equals -1, so:
h(15) = 9 * (-1) + 12
h(15) = -9 + 12
h(15) = 3
Therefore, if the riders get on at 15 seconds, the height of the Ferris wheel is 3 meters.