Mth163
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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Assignment # 11 - 2nd `qa program.I have copied question below:
Question: `qIf the time per swing is a x ^ .5, for the value determined previously for the parameter a, then what expression represents the number of swings per minute? How does this expression compare with the function you obtained for the number of swings per minute vs. length?
Your solution:
The time per swing is y = a x^5 and # of swings per minute = 60 / y then your expression would be represented by function = 60 / (a x*-5)
I'm also including your answer and my self critique:
Given Solution:
** Time per swing turns out to be a x^.5--this is what you would obtain if you did the experiment very accurately and correctly determined the power function. For x in feet a will be about 1.1.
Since the number of swings per minute is 60/(time per swing), you have f = 60 / (a x^.5), where f is frequency in swings / minute.
Simplifying this gives f = (60 / a) * x^.5.
This should be (60 / a) * x^-.5
60/a is just a constant, so the above expression is of form f = k * x^-.5, consistent with earlier statements.
60 / a = 60 / 1.1 = 55, approx., confirming our frequency model F = 55 x^-.5. **
Self-critique (if necessary):
ok but I still don’t under stand or I’m just confusing my self on this whole concept and with different data points I have nothing to compare to, and where did you get the 1.1 data from?
If you count cycles of a pendulum and make a table of time per swing vs. length in feet, you find that y = a x^.5 is the function that yields a relatively constant value of a.
You find this by substituting the various values of time per swing for y, the corresponding value of length in feet for x, and solving for a.
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In my self critique I explain what I don't understand, I am also including my data so you can see where I am getting my information.
x = length in inches - 33, 29, 25, 21, 17, 13, 9 and y = # of swings - 23, 22, 18, 16, 14, 12, 10
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not understanding this results in not finishing the rest of the assignment. I struggle with the geometry aspect of math and may need to seek some tutoring if unable to grasp what you are trying to explain in this program.
The number of swings doesn't decrease with length, it increases. So your data aren't quite correct.
Converting your x values to feet and using typical values of time per swing we get
x y
2.8 1.8
2.4 1.7
2.1 1.5
1.8 1.4
1.1 1.1
.8 .9
If we calculate a = y / x^.5 for each row of the table we get
a = 1.8 / sqrt(2.8) = 1.1
a = 1.7 / sqrt(2.4) = 1.1
etc.
Most results are close to 1.1.
This confirms that y = a x^.5 is a representative model for this data.
If you did the same thing with y = a x^2, or y = a x^-.5, your values of a would be nowhere near constant.
The corresponding frequency table might be
x y
2.8 33
2.4 35
2.1 40
1.8 45
1.1 52
.8 66
The model y = a x^.5 wouldn't work here, but the model y = a x ^-.5 would, and the values of a would be close to 55.