1. Given: A machine delivers constant power $P$ to a box moving on a horizontal straight line.

2. Use the relation between power and force: $P = Fv$ where $F$ is the pulling force and $v$ is the speed.

3. Relate force to acceleration: $F = m\frac{dv}{dt}$ So, $P = m\frac{dv}{dt} \cdot v$

4. Rearrange the equation: $m v \frac{dv}{dt} = P$ $v\,dv = \frac{P}{m}\,dt$

5. Integrate using the initial condition that the box starts from rest, i.e. $v=0$ at $t=0$: $\int_0^v v\,dv = \frac{P}{m}\int_0^t dt$ $\frac{v^2}{2} = \frac{P}{m}t$ $v = \sqrt{\frac{2Pt}{m}}$

   Hence, $v \propto t^{1/2}$

6. Now find displacement $x$: $\frac{dx}{dt} = v \propto t^{1/2}$ Therefore, $x \propto \int t^{1/2} dt \propto t^{3/2}$

7. Match with options: $x \propto t^{3/2}$ So the correct option is B.

Final Answer: $\boxed{t^{3/2}}$