Find the coordinates of the point(s) where the line <t, 4t+256, 8t> intersects the surface y=z^2+4x.
Setting y=z^2+4x, 4t+256=64t^2+4t. Cancelling out the 4t terms and simplifying one gets 4=t^2. The solutions are t=2 and t=-2. For t=2 one gets the point <2, 264, 16> and for t=-2 one gets the point <-2, 248, -16>.
Explain how a scientist can use spectra and redshift to determine if a body is moving towards or away from us.
Redshift is the phenomenon whereby electromagnetic wave wavelengths are changed. This change can occur gravitationally but also from the motion of an object due to the Doppler effect. It is helpful to think of this effect in terms of an ambulance. When the ambulance is moving towards you a higher pitch is heard. This is because the ambulance is moving towards the sound waves it is emitting and the peaks and troughs of the wave are closer together (lower wavelength) because they are being compressed. As the ambulance moves away from the waves it is creating the peaks and troughs are farther apart and stretched (higher wavelength) and a lower pitch is observed. This same concept is used when studying the motion of celestial bodies. If we look at a spectrum from a planet and see a characteristic emission line width for hydrogen but it is shifted to a higher wavelength then we say it is redshifted (because red is at the higher wavelength side of the visible spectrum) and thus moving away from us. Similarly if the spectrum is shifted to a lower wavelength we say it is blueshifted and moving towards us.
Explain why the Earth's rotation causes one to feel gravity's influence slightly less at the equator.
Points on a rotating body have constantly changing directions (and thus velocities), thus they have an acceleration. This acceleration is in a different direction as the gravitational acceleration so it deducts from it. This is known as the centrifugal force, one that is a "fictitious force" arising from being in a non-inertial reference frame (one that is accelerating). One can think of this situation in terms of Newton's law stating that objects will continue in a certain direction unless acted on by a force. A point in a rotating body at one instant in time has a velocity vector that wants to remain pointed in a specific direction. However, the rotating body changes this vector and a force is needed to do this. The gravitational force is different than this and (simply put) effectively remains pointed towards the center of the Earth.