example

advertisement
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
 10
b
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
Download
Related flashcards

Polynomials

21 cards

Ring theory

15 cards

Category theory

14 cards

Elementary algebra

12 cards

Representation theory

13 cards

Create Flashcards