9-A particle moves along a curve so that dx/dt=cos(t^2)...

Question

9-A particle moves along a curve so that dx/dt=cos(t^2) and dy/dt=e^sin(t). For what value of t,( 0 less than t which is less than 1), does the line tangent to the curve have of slope of 2

Answer 1

Given:

dx/dt = cos(t²)

dy/dt = e^sin(t)

For 0 < t ≤ 1 Alternatively written: t: (0, 1]

Find:

For dy / dx = 2, t = ?

Solution:

dy/dx = ( dy/dt ) / ( dx/dt )

dy/dx = e^sin(t) / cos(t²)

2 = e^sin(t) / cos(t²)

2 cos( t² ) = e^sin(t)

I am really unsure how to do the actual algebra manipulation to find the numerical values for t, but after plotting:

f(x) = 2 cos( x² )

g(x) = e^{sin( x )}

And after dragging around and zooming in until I found their intersections, between 0 and 1, I found that:

(For the graph x = ) = (For your problem) t = 0.647 s

I hope this helps! Message me in the comments if you have any questions, comments, or concerns!