<PRE><B>Question 1203200</B>
&lt;br&gt;
The term in the statement of the problem is &quot;inversely proportional&quot;, not &quot;indirectly proportional&quot;.&lt;br&gt;
There is no &quot;sum of two parts&quot;. The language in your textbook is misleading; using the word &quot;sum&quot; is a poor choice.&lt;br&gt;
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:&lt;br&gt;
{{{f=k(H)(sqrt(P))(1/D^2)}}} or {{{f=k((H*sqrt(P)/D^2))}}}&lt;br&gt;
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.&lt;br&gt;
Unfortunately, the numbers in the problem are not &quot;nice&quot;, so it is not a very good example for a student who is just learning the topic.&lt;br&gt;
Basic method....&lt;br&gt;
Find k using the given information&lt;br&gt;
{{{f=k((H*sqrt(P)/D^2))}}}&lt;br&gt;
Given: D = 8, H=40, P=1000, F=12&lt;br&gt;
Substitute to find k&lt;br&gt;
{{{12=k((40*sqrt(1000)/64))}}}&lt;br&gt;
{{{k=(12*64)/(40*sqrt(1000))=768/(400*sqrt(10))=(48/(25*sqrt(10)))}}}&lt;br&gt;
Solve the problem using this value of k and the new values of f, H, and D.&lt;br&gt;
{{{8=(48/(25*sqrt(10)))((30*sqrt(P)/100))}}}&lt;br&gt;
{{{sqrt(P/10)=(8)(25/48)(10/3)=125/9}}}&lt;br&gt;
{{{P/10=15625/81}}}&lt;br&gt;
{{{P=156250/81}}}&lt;br&gt;
ANSWER: P=156250/81&lt;br&gt;
Solving without finding k....&lt;br&gt;
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 &quot;nice&quot;.  But with this problem the ugly numbers make the work nearly as complicated.&lt;br&gt;
Solve the equation for P...&lt;br&gt;
{{{f=k((H*sqrt(P)/D^2))}}}&lt;br&gt;
{{{sqrt(P)=(f*D^2)/kH}}}
{{{P=((f*D^2)/kH)^2}}}&lt;br&gt;
Here we don&#39;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.&lt;br&gt;
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.&lt;br&gt;
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.&lt;br&gt;
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.&lt;br&gt;
The new value of P is then the old value of P, multiplied by all of these factors.&lt;br&gt;
{{{P=(1000)(4/9)(625/256)(16/9)=156250/81}}}&lt;br&gt;
And again (of course) the answer is 156250/81.&lt;br&gt;
</PRE>