Calculator User Manual

256
Chapter 19: Applications
19APPS.DOC TI-86, Chap 19, US English Bob Fedorisko Revised: 02/13/01 2:41 PM Printed: 02/13/01 3:05 PM Page 256 of 1819APPS.DOC TI-86, Chap 19, US English Bob Fedorisko Revised: 02/13/01 2:41 PM Printed: 02/13/01 3:05 PM Page 256 of 18
Reservoir Problem
On the TI
-
86, you can use parametric graphing animation to solve a problem.
Consider a water reservoir with a height of 2 meters. You must install a small valve on the
side of the reservoir such that water spraying from the open valve hits the ground as far
away from the reservoir as possible. At what height should you install the valve to
maximize the length of the water stream when the valve is wide open?
Assume a full tank at time=0, no acceleration in the x direction, and no initial velocity in the
y direction. Also, ignore valve-size and valve-type factors. Integrating the definition of
acceleration in both the x and y directions twice yields the equations x=v
0
t and
y=h
0
N
(gt
2
)
à
2. Solving Bernoulli’s equation for v
0
and substituting into v
0
t results in this pair
of parametric equations:
xt=t
‡
(2g(2
N
h
0
)) yt=h
0
N
(gt
2
)
à
2
t = time in seconds
h
0
= height of the valve in meters
g = the built-in acceleration of gravity constant
When you graph these equations on the TI
-
86, the y-axis (x=0) is the side of the reservoir
where the valve is to be installed. The x-axis (y=0) is the ground. Each plotted parametric
equation represents the water stream when the valve is at each of several heights.