Question

The position vector of an object that is moving in the xy-plane is (t+cost, t^3-75t) for t on [0, 2pi]. At which value of y does the object's path have a horizontal tangent?

I think this is the value at which dy/dt=0, or at t=5. At t=5, y= -250.

Answer 1

x = t + cos t
y = t^3 - 75 t

I think I want dy/dx = 0 not dy/dt = 0 but it amounts to the same here
one way
dy = (3 t^2- 75) dt
dx = (1 - sin t)
when is dy = 0 ?
3 t^2 = 75
t^2 = 25
t = +/- 5, yes use t = 5
then y = t^3-75t = 125 - 375 = - 250 agree

Answer 2

x = t + cos t
y = t^3 - 75 t

dy/dx = (dy/dt)/(dx/dt) = (3t^2-75)/(1-sint)
As Damon said, finding where dy/dx=0 is the same as finding where dy/dt=0, since the denominator is only zero when t=pi/2.