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(B) The maximum mass $m_{\max}$ depends on $v$, $g$, and $\rho$. The dimensional formula for mass is $[M^1 L^0 T^0]$.The dimensions of the variables a

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$6, -3, 1$

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(B) The maximum mass $m_{\max}$ depends on $v$, $g$, and $\rho$. The dimensional formula for mass is $[M^1 L^0 T^0]$.The dimensions of the variables are:$[v] = [L T^{-1}]$$[g] = [L T^{-2}]$$[\rho] = [M L^{-3}]$Given $m_{\max} = k v^x g^y \rho^z$, we write the dimensional equation:$[M^1 L^0 T^0] = [L T^{-1}]^x [L T^{-2}]^y [M L^{-3}]^z$$[M^1 L^0 T^0] = [M^z L^{x+y-3z} T^{-x-2y}]$Comparing the powers of $M, L,$ and $T$ on both sides:For $M$: $z = 1$For $T$: $-x - 2y = 0 \Rightarrow x = -2y$For $L$: $x + y - 3z = 0$Substituting $x = -2y$ and $z = 1$ into the $L$ equation:$-2y + y - 3(1) = 0$$-y = 3 \Rightarrow y = -3$Now, $x = -2(-3) = 6$Thus, the values are $x = 6, y = -3, z = 1$.

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