Question

Establishe dimensionally the relation between the reverberation time t of a hall,its surface area a,volume v and speed of sound v.given t is inversely proportional to v

Answer 1

Explanation:

I'm not sure entirely what you mean by a "dimensional relation" here, but in general we cannot take an arbitrary function of anything other than a dimensionless parameter because functions have Taylor expansions which sum together arbitrary powers of their arguments. Dimensional analysis therefore restrains you when you're about to say, "This variable is some unknown arbitrary function of these parameters": it says, "no, that function must have a very special form, so that the units work out to something other than nonsense." And the first thing it says is that the only truly-arbitrary functions involve dimensionless parameters. So, the first thing to look at is whether there are any dimensionless parameters in your case, and there is, because

[[V2]]=l6=[[A3]].[[V2]]=l6=[[A3]].

This means that something with units of tt will have the form

T=A1/2C f(V2/A3)T=A1/2C f(V2/A3)

for some arbitrary function ff.

What you've run into with the 3a+2b=13a+2b=1 equation is the arbitrariness of f. If you'll pardon the pun, ff is ineffable, we can't know it. We don't even have a form for it: sin(V/A3/2)+cos(V2/A3)sin⁡(V/A3/2)+cos⁡(V2/A3) is perfectly acceptable as far as dimensional analysis is concerned!

However, maybe "a dimensional relation" means that f(x)f(x) must take the form of kxnkxn for one kk and nn. Even then, the criterion you gave ("increases with V, decreases with A") only states that n>1/6n>1/6, which is not enough to specify