1. **Problem (a):** Given the displacement $x = Bt^2$, find the dimensions of $B$. 2. **Step 1:** Recall that displacement $x$ has dimensions of length $[L]$. 3. **Step 2:** Time $t$ has dimensions $[T]$, so $t^2$ has dimensions $[T^2]$. 4. **Step 3:** The equation is $x = Bt^2$, so dimensions satisfy: $$[L] = [B][T^2]$$ 5. **Step 4:** Solve for $[B]$: $$[B] = \frac{[L]}{[T^2]} = L / T^2$$ 6. **Answer (a):** The dimensions of $B$ are $L / T^2$. 7. **Problem (b):** Given $x = A \sin(2\pi ft)$, find the dimensions of $A$. 8. **Step 1:** The argument of sine must be dimensionless, so $2\pi ft$ is dimensionless. 9. **Step 2:** Since $t$ has dimensions $[T]$, frequency $f$ must have dimensions $1/T$ to make $ft$ dimensionless. 10. **Step 3:** Displacement $x$ has dimensions $[L]$, and since sine is dimensionless, $A$ must have the same dimensions as $x$. 11. **Answer (b):** The dimensions of $A$ are $L$. **Summary:** - (a) $[B] = L / T^2$ - (b) $[A] = L$