MTH 163
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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I keep running into problems with my modeling process. Somehow I end up with a model that works for only a couple of the given points and the rest don't come close to what they should be. This happened with the hot potato worksheet and now it's happening again with the completion of flow model worksheet. I believe I understand the process correctly, but my math seems to be wrong somehow. Can you help me? (see pasted work below)
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I believe I understand the modeling process correctly, but my math seems to be wrong somehow. (see pasted work below)
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The following are simulated depth vs. clock time data for the flow experiment:
clock t 3.6 7.2 10.8 14.4 18 21.6
depth 75.2 70.4 65.2 61.2 57.7 55.4
Clock times are in seconds, depths in cm.
Using a graph and simulaneous linear equations find a model for this data set and compare your model with the data.
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Where depth = y and time = x, we use the formula y = a x^2 + b x + c. Choosing 3 points to represent our data:
(3.6, 75.2)
(10.8, 65.2)
(21.6, 55.4)
... we may begin the modeling process.
First, we substitute our data into the quadratic equation:
75.2 = a * 3.6^2 + b * 3.6 + c or 12.96a + 3.6b + c = 75.2
65.2 = a * 10.8^2 + b * 10.8 + c or 116.64a + 10.8b + c = 65.2
55.4 = a * 21.6^2 + b * 21.6 + c or 466.56a + 21.6b + c = 55.4
Next, we eliminate the variable c by adding equations together:
116.64a + 10.8b + c = 65.2
-12.96a + 3.6b + c = 75.2
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103.68a + 7.2b = -10
466.56a + 21.6b + c = 55.4
-12.96a + 3.6b + c = 75.2
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453.6a + 18b = -19.8
Using these two equations, we eleminate b to solve for a:
103.68a + 7.2b = -10 (*18)
453.6 a + 18 b = -19.8 (*7.2)
1866.24a + 129.6b = -180
- 3265.92a + 129.6b = 142.56
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-1399.68a = -322.56
At this step you probably mean
1866.24a + 129.6b = -180
- (3265.92a + 129.6b = 142.56),
which would yield the result you obtained. However the right-hand side of the equation 453.6 a + 18 b = -19.8 has a sign opposite the coefficients on the left, and this doesn't appear to be the case with your equation.
You should probably get
-1399.68a = -37, approximately.
a = 4.339
We may now substitute a into the previous equation to determine b:
103.68(4.339) + 7.2b = -10
449.868 + 7.2b = -10
7.2b = -459.868
b = -63.871
Now with values for parameters a and b, we can solve for c using one of the original 3 quadratic equations:
12.96(4.339) + 3.6(-63.871) + c = 75.2
56.233 - 229.937 + c = 75.2
-173.704 + c = 75.2
c = 248.904
Check our work by using a second of the original formulas:
116.64(4.339) + 10.8(-63.871) + c = 65.2
506.101 - 689.807 + c = 65.2
-183.706 + c = 65.2
c = 248.906
We now have the model for depth vs time:
y = 4.339a^2 -63.871b + 248.906
Let's test this with one of the given points, (3.6, 75.2), substituting 3.6 for x:
y = 4.339 * 3.6^2 - 63.871 * 3.6 + 248.906
y = 56.233 - 229.9356 + 248.906
y = 75.2034
14.4, 61.2
y = 4.339 * 14.4^2 - 63.871 * 14.4 + 248.906
y = 899.735 - 919.742 + 248.906
y = 228.899
See my note and see if you agree. You're following the process correctly, but it seems you have a sign error (very common in problems of this nature).