The discussion centers on the calculation of the rate at which an ant's shadow moves on the y-axis, given its position and movement rates. The user initially calculated the shadow's movement rate as -9/16 but faced criticism for potential errors in their differentiation process. The correct approach involves using similar triangles to relate the shadow length and the ant's coordinates, leading to the conclusion that the correct rate is indeed -9/16, despite the user's confusion over signs in their calculations.
PREREQUISITESStudents studying calculus, particularly those focusing on related rates, as well as educators seeking to clarify concepts of shadow movement and differentiation techniques.
You "lost a sign"mit_hacker said:Let the coordinates of the ant be x,y. Let the length of the shadow be s.
Our goal is to find ds/dt. By similar triangles, s/y = 5/(5-x).
So, 5s - sx = 5y
Differentiating,
5(ds/dt) - (s(dx/dt) + x(ds/dt)) = 5(dy/dt)
When (x,y) = (1,2), s = 5/2.
5(ds/dt) - (5/2)(1/2) - (1)(ds/st) = 5(-1/5).
4(ds/dt) = -9/4.
ds/st = -9/16