## anonymous 4 years ago CHALLENGE: The equation of motion for a weight suspended from a particular spring is d( t)=5sin4t-2cos4t, where d is the displacement from the equilibrium position in centimetres and t is the time elapsed in seconds. How many time does the weight pass through the equilibrium position in the first two cycles?

1. anonymous

equation: $d(t)=5\sin4t-2\cos4t$

2. anonymous

help meeeeeeeeeeee :'(

3. dumbcow

well you would have to solve for when d(t) = 0 --> 5sin(4t) = 2cos(4t) Note: sin/cos = tan --> tan(4t) = 2/5 --> 4t = arctan(2/5) = .3805 t = 0.095 Now 2 cycles means 2 periods, Period = 2pi/4 = pi/2, 2*pi/2 = pi --> 0 < t < pi/4 or 0< t < .785

4. anonymous

YES, this is the step that i was looking for. I totally forgot about going from sin and cos to tan by division. Thanks a lot for this!

5. anonymous

so the final answer is 1 because t only equals 0.095 in range of 0<t<0.785? just making sure.

6. dumbcow

yeah thats what i get :)

7. anonymous

ok thanks!