PROBLEM
High School Pre-Cal
While watching a waterwheel turn in a river, Jenny noticed that, as the wheel turned, the distance from a point on its edge to the water varied sinusoidally with time. Jenny started her stopwatch when a particular point, M,was at its highest point, 17 feet above the water.
If the diameter of the waterwheel was 20 feet and it completed one turn every 12 seconds, what did Jenny's stopwatch read when point M next touched the water?
A.about 4.5 seconds
B.about 5.1 seconds
C.about 5.8 seconds
D.about 6.3 seconds
While watching a waterwheel turn in a river, Jenny noticed that, as the wheel turned, the distance from a point on its edge to the water varied sinusoidally with time. Jenny started her stopwatch when a particular point, M,was at its highest point, 17 feet above the water.
If the diameter of the waterwheel was 20 feet and it completed one turn every 12 seconds, what did Jenny's stopwatch read when point M next touched the water?
A.about 4.5 seconds
B.about 5.1 seconds
C.about 5.8 seconds
D.about 6.3 seconds
SOLUTION
Looking at the following diagram, the waterwheel moves from the top position of A to the bottom position of C at the water line.
* * * * * * * * * * A * * * * * * * * * *
* * * * * * m * * * * * * * m * * * * * *
* * * * m * * * * * * * * * * * m * * * *
* * * m * * * * * * * * * * * * * m * * *
* * m * * * * * * * * * * * * * * * m * *
* * * * * * * * * * * * * * * * * * * * *
* m * * * * * * * * * * * * * * * * * m *
* * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * *
D * * * * * * * * * O – – – – – – F – – B
* * * * * * * * * * | * \ * * * * ¦ * * *
* * * * * * * * * * | * * \ * * * ¦ * * *
* * * * * * * * * * | * * * \ * * ¦ * * *
* m * * * * * * * * | * * * * \ * ¦ * m *
* * * * * * * * * * | * * * * * \ ¦ * * *
* * m * * * * * * * | * * * * * * \ m * *
∼ ∼ ∼ m ∼ ∼ ∼ ∼ ∼ ∼ E ∼ ∼ ∼ ∼ ∼ ∼ C ∼ ∼ ∼
* * * * m * * * * * * * * * * * m * * * *
* * * * * * m * * * * * * * m * * * * * *
* * * * * * * * * * m * * * * * * * * * *
Period, T = 12 s
Diameter, d = 20 ft
Radius, r = 10 ft = length of OC
length of OE, distance from center to water line = 7 ft
Using trigometry, m∠BOC = sin-1(OC/OE) = sin-1(7/10) = 44.43°
Therefore, arc ABC = 134.43°
If it takes the waterwheel 12 s to complete one revolution, then the time to revolve from point A to point C is (134.43°/360°)(12 s) ≈ 4.48 s
The time of 4.48 s is closest to answer choice A.about 4.5 seconds.
