math
moody-
water_works_extra_credit_ver_calculus_fa.pdf
MAT 2280 – Water Flow Approximation Integration Sec. 7.7
Just beyond Sinclair across Robert
Drive there is a circular water pipe that
dumps rain water off the street into the
river. Use the data collected on a rainy
day in June to determine how many
gallons of water was dumped into the
river from 7:00 AM to 7:00 PM.
The values in the table below were calculated from
flow rate and water depth measurements taken at
various times of the day starting at 7:00 AM.
PART 1 Preparing the Raw Measurement Data
Example: To calculate the gallons/hour at 1:00 PM the following measurements had to be taken.
The diameter of the pipe was measured at 65.25 inches. The depth of the water was 8.25 inches
and the average flow rate was 1.8 meters/second. How was 270,662.00 gal/hr calculated?
Time HOUR gal/sec gal/hr
7:00 AM 0 20.01 72036.00
10:00 AM 3 55.80 200880.00
1:00PM 6 75.18 270662.00
4:00 PM 9 65.10 234360.00
7:00 PM 12 55.98 201528.00
Diameter = 65.25 inches Depth = 8.25 inches Flow-rate Probe = 1.8 m/s
A Flow rate Probe for Vernier’s
LabPro was used to collect data to
the TI-86 graphing calculator
PART 2 Numerical Methods
1) Use rectangles to estimate the total number of gallons of water dumped into the river from
7:00 AM to 7:00 PM.
2) Use trapezoids to estimate the total number of gallons of water dumped into the river from
7:00 AM to 7:00 PM.
3) Use Simpson’s Rule to estimate the total number of gallons of water dumped into the river
from 7:00 AM to 7:00 PM.
0.00
50000.00
100000.00
150000.00
200000.00
250000.00
300000.00
0 3 6 9 12 15
W a
te r
F lo
w (
g a
l/ h
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Time (hours)
Water Flow
Time HOUR gal/sec gal/hr
7:00 AM 0 20.01 72036.00
10:00 AM 3 55.80 200880.00
1:00PM 6 75.18 270662.00
4:00 PM 9 65.10 234360.00
7:00 PM 12 55.98 201528.00
PART 3 Algebraic Function Model
4 3 2
( ) 80.544 1740.07 7305.1 34518.6 72036f t t t t t
1) The rain began about 6:30 AM and continued till about 8:30 when it began to taper off. At
2:30 PM it began to rain again but not as heavy. Explain why the graph appears the way it is.
Comment on extreme values and possible inflection points and what they mean etc…
2) What is a practical domain and range of this function?
3) Use the function to estimate the number of gallons dumped from 7:00 AM to 7:00 PM.
Do this by hand, integrating the function with the appropriate limits.
Total Gallons from 7:00 AM to 7:00 PM =
12
0
( ) 2, 545, 370.84f t dt gal
On calculator: fnInt(Y1,X,0,12) = 2,545,370.84 (Note: I used “t” for the time “independent
variable” but for convenience I used X for variable in the TI-84.)
4) How many gallons of water were dumped into the river between 9 AM and 5:00 PM?
5) Compare the approximation methods to the value obtained be the above integral. Which
method(s) do you think work better and why?
