Question 1203200: I have another question about proportion.
THE TEXTBOOK GIVES THE FOLLOWING QUESTION under the heading ‘variation as the sum of two parts’; I cannot change the wording.
‘For a certain series of experiments it is known that a quantity F is directly proportional to H and the square root of P and inversely proportional to the square of D.
If D = 8, H=40 and P=1000 when F=12, calculate the value of P when
F = 8, d = 10 and H=30’.
The problem I seem to be having here is how to set up the equation. The book is very vague about the difference between joint variation (which I have no problem with) and variation as the sum of two parts, but I am assuming, following the example in the book, that sum means add.
The notation that I am using here is the notation in the book. If anyone finds it bizarre, there is nothing I can do about that.
I assume then that the equation could be stated thus: obviously I am using “Word” so I have included notes so that people are not in any doubt about the meanings.
F∝H - F is directly proportional to H;
F∝√P - F is directly proportional to the square root of P;
F∝1/D2 - F is indirectly proportional to the square of D.
I re - wrote this thus:
F = kH;
F = k√P;
F = k(1/D2)
WHERE K STANDS FOR THE CONSTANT OF PROPORTIONALITY.
I then combined the terms thus:
F=k(H*√P)+(1/D2)
I have added 1/D2 because the title of this section of the book that I am using is the sum of two parts, so I am assuming that 1/D2 is the second part and, thus, should be added.
I cannot get the correct answer to the question even if I multiply all of the terms and I cannot see where I am going wrong. I have tried to find information about “Variation as the sum of two parts” on the internet, and “Bing”, Google” and “You Tube” cannot understand the difference between the words variation and variance, which does not surprise me in the least, so no help there. The solution is probably very simple, but I just cannot see it
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! ------------------------------------------------------------------------------------------------------------
a quantity F is directly proportional to H and the square root of P and inversely proportional to the square of D.
If D = 8, H=40 and P=1000 when F=12, calculate the value of P when
F = 8, d = 10 and H=30’.
--------------------------------------------------------------------------------------------------------------
You were next given some value so you can determine value of k.
,
,
Answer by ikleyn(52780)  (Show Source):
You can put this solution on YOUR website! .
I do not think that "indirectly proportional" is a correct Math term.
I do not think that "indirectly proportional" is a Math term, at all.
There is commonly accepted term "inversely proportional" in Math,
but I NEVER saw the term "indirectly proportional" in peer reviewed Math textbooks.
When I come with the term "indirectly proportional" to Google, it shows the results
for "inversely proportional" only and instead.
Could you give a precise reference to your textbook (its name, author, ISBN number, year of publication, publishing company).
I doubt that it is a normal/regular printed peer reviewed Math textbook.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The term in the statement of the problem is "inversely proportional", not "indirectly proportional".
There is no "sum of two parts". The language in your textbook is misleading; using the word "sum" is a poor choice.
The variation is directly proportional to H AND directly proportional to the square root of P AND inversely proportional to the square of D. All of those parts are multiplicative factors:
 or 
There are two basic ways to solve the problem. If you want to learn about direct and inverse variation, then use the given information to determine the proportionality constant k and then use it to solve the problem. For more advanced students, it is possible to solve the problem without finding k.
Unfortunately, the numbers in the problem are not "nice", so it is not a very good example for a student who is just learning the topic.
Basic method....
Find k using the given information
Given: D = 8, H=40, P=1000, F=12
Substitute to find k
Solve the problem using this value of k and the new values of f, H, and D.
ANSWER: P=156250/81
Solving without finding k....
Solving the problem by seeing how the change in each parameter affects the final value is much faster and easier than finding the value of k if the numbers in the problem are "nice". But with this problem the ugly numbers make the work nearly as complicated.
Solve the equation for P...
Here we don't need the constant of variation; we are only going to see how the value of P changes for each of the changes in parameters f, D, and H.
The value of f changes from 12 to 8, a ratio of 2/3; the variation is direct, so that causes the value of P to change by a factor of (2/3)^2 = 4/9.
The value of D changes from 8 to 10, a ratio of 5/4; the variation is direct, so that causes the value of P to change by (5/4)^4 = 625/256.
The value of H changes from 40 to 30, a ratio of 3/4; the variation is inverse, so that causes the value of P to change by (4/3)^2 = 16/9.
The new value of P is then the old value of P, multiplied by all of these factors.
And again (of course) the answer is 156250/81.
|
|
|
