Help me please! Thank you so much

Question

A weight on a spring is oscillating sinusoidally. We know that the weight reaches its highest value for the first time at t = 2 seconds at a height of 20 inches. If the next time it reaches its highest point is at t= 12 seconds, find:

a.The sinusoidal equation for the height of this weight at any time t if the minimum height of the weight is 4 inches.

b.Find the height of the spring at t = 15.5 seconds

c.Find the first THREE times that the weight is 18 inches off of the ground.

Answer 1

The height varies from. 4-20 which puts the midline at 12 and amplitude of 8. since the graph starts at a maximum at t=2 this is a cosine graph shifted right by 2

period is 12-2 or 10 B=2pi/b=2pi/10 =Pi/5 so. Equation is

y= 8[cos(pi/5)(t-2)] +12

to find the height at.t=15.5 solve y=8cos[(pi/5)(15.5-2)] +12

y= 8cos(pi/5)(13.5) +12

y=7.3seconds

to find the times when height is 18 solve 18=8cos[pi/5)(x-2)]+12

6= 8cos[pi/5(x-2) ]

.75 = cos(pi/5)(x-2). Take inverse cosine of both sides

.7227=(pi/5)(x-2)

x-2 =1.1 x=3.1sec which is 1.1 se from the start so. Next time it will be 18 is 12-1.1= 10.9sec and 3.1 + one period or 3.1 +10 = 13.1 sec