Question number eight. A rectangular sheet of paper was first rolled into a cylinder and then cut along a plane passing through points A and B. The lower part of the sheet was then unrolled. Which picture represents this part of the sheet? I assigned a coordinate system to our cylinder, so we have the positive x-axis, the positive y-axis, the positive z-axis, and the points A and B. I will suppose that A lies on the positive x-axis, or right above it, and that the cylinder has radius r, so A has coordinates r, 0, and then let's say it's A units above the xy-plane, and similarly B will be on the negative x-axis, B units above the xy-plane, and then to move around the red ellipse, we will assign the cylindrical coordinate system to our space, so we will let r be r cosine of theta, that will be x, r sine of theta, that will be y, and z will be z. So then looking in the zx-plane, we can find an equation of a line passing through the points A and B, and that z will determine for us the distance from a point on the ellipse to the xy-plane, and it will be a function of theta. As we revolve here through an angle theta, that is at least zero, really we should say that theta belongs to the interval from negative pi to pi as we make one revolution, then z will essentially describe for us the shape of the top boundary of each of these figures we have in our multiple choice. So let's get that equation of the line passing through A and B using the point slope formula. We have z minus A, so I'm using the point A here, is equal to the slope, which will be the change in z over the change in x, x minus r, and that is z is equal to, the slope is now A minus B over 2r, that's r minus negative r, x minus r plus A, and we can simplify this a little bit more, this will be A minus B over 2r times x, plus B minus A over 2, plus A like this, so that is finally A minus B over 2r times x, plus B plus A all over 2, and we see that with our coordinate system here, x is really equal to r times the cosine of theta, so finally this all simplifies to some constant, let's call it alpha, times the cosine of theta, plus another constant, let's call it beta. So our graph, z, as a function of theta, looks like the graph of the cosine function, and the only bounding curve here that looks anything like a periodic graph of the cosine function, or at least a piece of it, if we only restrict our attention to theta ranging from negative pi to positive pi, is the graph in C, so that must be the answer.

cylindrical coordinates

cosine wave

boundary line

cylinder

shape determination