1. Given: The machine delivers constant power $P$ to move a body along a straight line.

2. Use the definition of power: $P = \frac{dW}{dt}$ Since power is constant, $W = Pt$

3. Relate work to kinetic energy: Assuming the body starts from rest and all the work goes into increasing kinetic energy, $Pt = \frac{1}{2}mv^2$

4. Find how velocity depends on time: $v^2 \propto t \quad \Rightarrow \quad v \propto t^{1/2}$

5. Now relate distance and velocity: $v = \frac{dx}{dt} \propto t^{1/2}$ So, $dx \propto t^{1/2}dt$

6. Integrate: $x \propto \int t^{1/2}dt = \frac{2}{3}t^{3/2}$

7. Therefore: $x \propto t^{3/2}$

8. Option check:

   • A: $t^{3/4}$ — incorrect
   • B: $t^{3/2}$ — correct
   • C: $t^{1/4}$ — incorrect
   • D: $t^{1/2}$ — incorrect

Hence, the distance moved is proportional to $\boxed{t^{3/2}}$.