I can’t seem to figure out the equations to solve this problem!

Space-time metric ds2= -dt2+at2dx2+bt2dy2+ct2dz2 (a)how does the area of a rectangle with vertices at ( x=0, y=0, z=0 ), (1,0,0 ), ( 0,1,0 ) and (1,1,0 ) changes with time? (b) if a photon moving in the x direction starts from the origin at t=1, when does it reach x=l? (ds=0 for the trajectory of a photon.) (c) the proper time for a particle is defined by cd=ds. what is the proper time for a particle to move along the trajectory ( x(t)=t, y(t)=0, z(t)=0 ) from t=0 to 1 ? what is the condition on the parameter a so that the velocity of the particle never exceeds the speed of light ?

A250-n box is at rest on a frictionless horizotal floor. then a person shoves the box with a non-zero force of magnitude p. which statement best describes the relationship betweeen p and the motion of the box after it was shoved? -the box will not move when p< 250n but otherwise will accelerate at a constant rate that is directly proportional to p - the box will begin moving at a speed that is directly proportional to p, then will gradually slow to a halt -the box will move with constant velocity at a speed that is directly proportional to p -box will accelerate across the floor at a constant rate that is directly proportional to p

Consider the growth of a 20-nm-diameter silicon nanowire onto a silicon wafer. the temperature of the wafer surface is maintained at 2400 k. assume the thermal conductivity of the silicon nanowire is 20 wm-1k-1 and all its surfaces including the tip are subjected to convection heat transfer with the coefficient h = 1×105 wm-2k-1 and t∞ = 8000 k. when the nanowire grows to l = 300 nm, what is the temperature of the nanowire tip (t (x =