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Mark Twain sat on the deck of a river
steamboat. As the paddlewheel
turned, a point on the paddle blade
moved in such a way that its distance
from the water’s surface was a
sinusoidal function of time. When his
stopwatch read 4 seconds, the point
was at its highest, 16 feet above the
water’s surface. The wheel’s diameter
was 18 feet, and it completed a
revolution every 10 seconds.
Sketch a graph of the sinusoid.
18 ft
y
4 sec
16 ft (max)
One revolution every 10 sec.
10
x
4
Height
(ft)
8
12
16
Time
(sec)
Write equation of the sinusoid.
Period: 10 seconds per cycle
2
b
 10

b
5
Phase shift: Maximum starts at 4 seconds
Vertical Shift: middle of graph (16 + -2)/2 = 7
Amplitude: distance from middle of graph 16 – 7 = 9
(no reflection if starting at max)
Therefore, the equation is: y  9 cos

5
( x  4)  7
How far above the surface was the point
when Mark’s stopwatch read 5 seconds?
17 seconds?
By using the equation:
y  9 cos

5
( x  4)  7
We can plug in 5 and 17 into the x and solve
for y (on calculator with trace or table)
When x = 5 sec , y = 14.3 ft
When x = 17 sec , y = 4.2 ft
What is the first positive value of time at
which the point was at the water’s surface?
Find the intersection of y = 0 (water’s surface) and
the sinusoid representing the paddlewheel.
Water’s surface