Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are Y1 and Y2. The smallest possible amplitude of the resultant wave is:

a) Y1 + Y2 and occurs if they are 180 degrees out of phase
b)[Y1 - Y2] and occurs if the are 180 degrees out of phase
c) Y1 + Y2 and occurs if they are in phase
d) [y1-Y2] and occurs if they are in phase
e) [Y1-Y2] and occurs if they are 90 out of phase

My thoughts:
All the explainations of resultants in my book for waves show Y1+Y2=Resultant for the theory of superposition. So I think then to get the smallest wave it would be A. Y1 + Y2 if they are 180 degrees out of phase.

The displacement of a string is given by
y(x,t)=Ysin(Kx + wt)
speed of the wave is
a)2pik/w
b)w/K
c)wk
d)2pi/k
e)k/2pi

Work:
v=lamba*f
k=2pi/lambda
w=2pi/T
v=2pi/lambda * 2pi/(1/f)
v=w/k
so I choose b

Help is appreciated :)

You are correct in your understanding that the smallest possible amplitude of the resultant wave is Y1 + Y2 when the two waves are 180 degrees out of phase. This is because when two waves are 180 degrees out of phase, the crests of one wave align with the troughs of the other wave, leading to cancellation and a smaller resultant amplitude.

For the second question about the speed of the wave, you correctly use the formulas for wave speed (v), wave number (k), angular frequency (w), and wavelength (lambda). The formula v = w/k relates the speed of the wave to its angular frequency and wave number. Since k = 2pi/lambda, you substitute this value for k into the equation and get v = w/(2pi/lambda). Simplifying this expression, you can rearrange to v = w * lambda / (2pi). From this equation, you can see that the correct answer is b) w/K, which matches your initial choice.

Therefore, your answers are:
1) a) Y1 + Y2 and occurs if they are 180 degrees out of phase, and
2) b) w/K