A gas bubble from an explosion under water oscillates with a period T proportional to pap "E where P is the statie pressure, p is the density of water and E is the total energy of the explosion. So, values of a, b and care ..... ... and ... respectively.

Explanation

Here `T prop p^(a) rho^(b) E^(c)` is given.
`:. T=kP^(a) rho^(b)E^(c)` where k is a dimensionless constant.
`[T]=T^(1),[K]=M^(0)L^(0)T^(-1)`
`:.[T]=[k][P]^(a)[rho]^(b) [E]^(c)`
But `[P]=M^(1)L^(-1)T^(-2)`
`:.T^(1)=(M^(1)L^(-1)T^(-2))^(a)(M^(1)+L^(-3))^(b)(M^(1)L^(2)T^(-2))^(c)`
`( :.[rho]=M^(1)L^(-3)T^(0)[E]=M^(1)L^(2)T^(-2))`
`:. T^(1)=M^(a+b+c)L^(-a-3b+2c)T^(-2a-2c)`
Comparing powers of T, we get
`-2a-2c=1:.a+c=-(1)/(2)...(i)`
Comparing powers of M, we get
`-a-3b+2c= :.2c=a+3b=a+(3)/(2)` [ From eq. (ii)
`:.2c-a=(3)/(2)...(iii)`
On addition of eq. (i) and (ii) `a+c=-(1)/(2)`
`(+-a+2c=(3)/(2))/(3c=1)`
`:.c=(1)/(3)...(iv)`
Putting value of c in eq (i) we get `a+(1)/(3)=-(1)/(3)`
`:. a=-(1)/(2)-(1)/(3)`
`:.a=-(5)/(6)...(v)`
From eq. (ii), (iv) and (v),
`a=-(5)/(6),b=(1)/(2) and c=(1)/(3)`