I did this work but it is wrong. Can you help me?

Question

A weight on a spring varies in height sinusoidally. If the highest point the spring reaches is at t = 3 seconds and a highest of 50 inches, and the lowest point occurs at t = 7 seconds at a height of 10 inches.

a. Find the equation of the height of the weight at any time t.

b. Find the height of the weight at t = 6 seconds.

c. What are the first two times that the spring is 23 inches off of the ground?

Answer 1

h(t) = Asin(B(t+C)) + D Assuming no dampening

D is the midpoint = (50+10)/2 = 30

The range of sin() is ±1, so

h(3) = A(1)+30 = 50 --> A = 20

h(7) = 20(-1)+30 = 10 checks out

B = 2π/Period and Period = 2(7-3)=8 --> B = 2π/8 = π/4 (because half cycle is from max to min)

For minimum and maximum

3π/4+Cπ/4 = π/2 --> C=-1

7π/4+Cπ/4 = 3π/2 --> C=-1

h(t) = 20sin(π/4(t-1)) + 30

b)

h(6) = 20sin(5π/4) + 30 = 20(-√2/2) + 30 = 15.86 inches

c)

20sin(π/4(t-1)) + 30 = 23

Solve for t

20sin(π/4(t-1)) = -7

π/4(t-1) = sin^-1(-7/20) = -0.1138π

t = 4(-0.1138) +1 = 0.5447 seconds First time

Second time:

20sin(π/4( 0.5447-1-x)) + 30 = 23

x = 5 seconds later, so t = 5.447 seconds