1. **Problem Statement:** Given the formula $$P = Ae^{ax + bt}$$ where $$P$$ is pressure, $$x$$ is displacement, $$t$$ is time, and $$A$$, $$a$$, and $$b$$ are constants, find the dimension of $$\frac{a}{b}$$. 2. **Understanding the formula:** The exponent in an exponential function must be dimensionless. Therefore, the quantity $$ax + bt$$ must be dimensionless. 3. **Analyzing dimensions:** Let the dimensions be: - $$[P] = M L^{-1} T^{-2}$$ (pressure dimension) - $$[x] = L$$ (displacement dimension) - $$[t] = T$$ (time dimension) Since $$ax + bt$$ is dimensionless, the dimensions of $$ax$$ and $$bt$$ must both be dimensionless. 4. **Dimension of $$a$$:** Since $$ax$$ is dimensionless, $$$[a][x] = 1 \implies [a] = [x]^{-1} = L^{-1}$$$ 5. **Dimension of $$b$$:** Since $$bt$$ is dimensionless, $$$[b][t] = 1 \implies [b] = [t]^{-1} = T^{-1}$$$ 6. **Dimension of $$\frac{a}{b}$$:** $$$\frac{[a]}{[b]} = \frac{L^{-1}}{T^{-1}} = L^{-1} T$$$ **Final answer:** The dimension of $$\frac{a}{b}$$ is $$L^{-1} T$$, which corresponds to option (3).