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

1

• 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 waters surface was a sinusoidal function of
  time. When his stopwatch read 4 seconds, the
  point was at its highest, 16 feet above the
  waters surface. The wheels diameter was 18
  feet, and it completed a revolution every 10
  seconds.

2
Sketch a graph of the sinusoid.

4 sec 16 ft (max) One revolution every 10
sec.
3
Write equation of the sinusoid.

• Period 10 seconds per cycle
• 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
4
How far above the surface was the point when
Marks stopwatch read 5 seconds? 17 seconds?

• By using the equation
• 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

5
What is the first positive value of time at
which the point was at the waters surface?
Find the intersection of y 0 (waters surface)
and the sinusoid representing the paddlewheel.
Waters surface