Answer:



Explanation:
Notation
In order to do the dimensional analysis we need to take in count that we need to conditions:
a) The energy A is released in a small place
b) The shock follows a spherical pattern
We can assume that the size of the explosion r is a function of the time t, and depends of A (energy), the time (t) and the density of the air is constant
.
And now we can solve the dimensional problem. We assume that L is for the distance T for the time and M for the mass.
[r]=L with r representing the radius
[A]=
A represent the energy and is defined as the mass times the velocity square, and the velocity is defined as 
[t]=T represent the time
represent the density.
Solution to the problem
And if we analyze the function for r we got this:
![[r]=L=[A]^x [\rho]^y [t]^z](https://tex.z-dn.net/?f=%5Br%5D%3DL%3D%5BA%5D%5Ex%20%5B%5Crho%5D%5Ey%20%5Bt%5D%5Ez%20)
And if we replpace the formulas for each on we got:
![[r]=L =(\frac{ML^2}{T^2})^x (\frac{M}{L^3})^y (T)^z](https://tex.z-dn.net/?f=%5Br%5D%3DL%20%3D%28%5Cfrac%7BML%5E2%7D%7BT%5E2%7D%29%5Ex%20%28%5Cfrac%7BM%7D%7BL%5E3%7D%29%5Ey%20%28T%29%5Ez%20)
And using algebra properties we can express this like that:
![[r]=L=M^{x+y} L^{2x-3y} T^{-2x+z}](https://tex.z-dn.net/?f=%5Br%5D%3DL%3DM%5E%7Bx%2By%7D%20L%5E%7B2x-3y%7D%20T%5E%7B-2x%2Bz%7D)
And on this case we can use the exponents to solve the values of x, y and z. We have the following system.

We can solve for x like this x=-y and replacing into quation 2 we got:



And then we can solve for x and we got:

And if we solve for z we got:

And now we can express the radius in terms of the dimensional analysis like this:

And K represent a constant in order to make the porportional relation and equality.
The problem says that we can assume the constant K=1.
And if we solve for the energy we got:


And now we can replace the values given. On this case t =0.025 s, the radius r =140 m, and the density is a constant assumed
, and replacing we got:

And we can convert this into ergs we got:

And then we know that 1 g of TNT have 
And we got:
