I am reading a book on calculus and I got to the part about trigonometry. There is an illustrative problem for the functions.

" The crank and connecting rod of a steam engine are three and ten feet long respectively, and the crank revolves at a uniform rate of 120 r.p.m. At what rate is the crosshead moving when the crank makes an angle of 45 degrees with the dead center line?"

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Now before I read the solution in the book I tried to solve it myself. I expressed the relation by which the crosshead moves as

x = r - r cos(angle)(in the picture what i call x is the distance from point A to the circle, they call x the distance from O to C),

because the crosshead is x away from the furthest right point it can go. At 90deg it is exactly r away from that point, at 180deg 2r and then back. I then differentiated this equation and I got

dx = d(r) - d(rcos(angle)),However this wasn't the right solution. In the book they give the solution that involves expressing the length from the center of the crank to the rod. So they then divide the resulting triangle into 2 right-angle triangles (OPA and PAC). Then x = OA + AC; x = r cos(angle) + sqrt(a

dx = 0 - r d(cos(angle)),

dx = r sin(angle) d(angle),

dx/dt = 3 sqrt(2)/2 4pi/sec,

dx/dt = 26.65ft/sec

^{2}sin^{2}(angle) + b^{2}). When this is differentiated to the end they get(they call radius a):

dx/dt=-a sin(angle)[1+ (cos(angle)/(sqrt(b/a)Now when you insert the known quantities you get^{2}- sin^{2}(angle))]d(angle)/dt

dx/dt = -32.44ft/sec(minus because it's moving left,mine was the opposite).

I don't see how the solutions can be different, or why the amount by which the crank moves at all depends on b. Shouldn't it only depend on the radius since the crank can't move more to the left or right than the raidus)?

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