Question 1172685
The pressure of the gas is directly proportional to the temperature and inversely proportional to
its volume.
A. Write the variation equation.
<pre>
For all proportion problems, start with this:

Varying           "directly" or product of "jointlys" or 1 if none 
quantity  = k × ----------------------------------------------------------
                inversely variable or product of "inverselys" or 1 if none

In this problem the varying quantity is P for pressure. The "directly" is T for
temperature.  We have one "inversely" variable, T for Volume.  We have no
"jointly" variable. So we have P on the left, T on the top of the fraction and
V on the bottom:

{{{matrix(1,3,P,""="",k*expr(T/V))}}}

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</pre>B. What will happen to the pressure if the volume is reduced to half and the 
temperature is doubled?<pre>
Let's substitute 2V for V in {{{matrix(1,3,P,""="",k*expr(T/V))}}}, and {{{expr(1/2)P}}} for P and simplify:

{{{matrix(1,3,Pressure,""="",k*expr((2T^"")/(expr(1/2)V^"")))}}}

We simplify the complicated term {{{k*expr((2T^""^"")/(expr(1/2)V))}}}{{{""=""}}}{{{k*expr((2T^"")/(expr(1/2)V))}}}{{{""*""}}}{{{2^""/2^""}}}{{{""=""}}}{{{k*expr((2T^"")/(expr(1/cross(2))V))}}}{{{""*""}}}{{{2^""/cross(2)^""}}}{{{""=""}}}{{{(4T)/V}}}

Since {{{Pressure=(4T)/V}}} is 4 times {{{P=T/V}}}, the answer to B is:

"The pressure will quadruple."

Edwin</pre>