1. **State the problem:** We have the volume $V$ as a function of time $t$ given by $$V = X + \frac{Y}{t} + Z t^4$$ We need to find the correct dimensional formulas for $X$, $Y$, and $Z$ assuming the equation is dimensionally correct. 2. **Recall the dimension of volume:** Volume has dimension of length cubed, i.e., $$[V] = L^3$$ 3. **Analyze each term:** Since the sum of terms must have the same dimension, each term must have dimension $L^3$. - For $X$, since it is added directly to $V$, its dimension must be $$[X] = L^3$$ - For $\frac{Y}{t}$, the dimension is $$\left[\frac{Y}{t}\right] = \frac{[Y]}{[t]} = L^3$$ Therefore, $$[Y] = L^3 \times [t] = L^3 T$$ - For $Z t^4$, the dimension is $$[Z t^4] = [Z] \times [t]^4 = L^3$$ Therefore, $$[Z] = \frac{L^3}{T^4} = L^3 T^{-4}$$ 4. **Match with options:** - $X = L^3$ - $Y = L^3 T$ - $Z = L^3 T^{-4}$ This corresponds to option **b**. **Final answer:** Option b: $X = L^3$, $Y = L^3 T$, $Z = L^3 T^{-4}$