Question

The equation models the height h in centimeters after t seconds of a weight attached to the end of a spring that has been stretched and then released.
a. Solve the equation for t.

b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position. Round your answers to the nearest hundredth.

c. Find the times at which the weight is at a height of 1 cm, of 3 cm, and of 5 cm below the rest position for the second time. Round your answers to the nearest hundredth.

Answer 1

This is the missing equation that models the hieght and is misssing in the question:

h= 7cos(π/3 t)


Answers:

a. Solve the equation for t.


1) Start: h= 7cos(π/3 t)


2) Divide by 7: (h/7) = cos(π/3 t)


3) Inverse function: arc cos (h/7) = π/3 t

4) t = 3 arccos(h/7) / π ← answer of part (a)



b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position. Round your answers to the nearest hundredth.

1) h = 1 cm ⇒ t = 3 arccos(1/7) / π

t = 1.36 s← answer


2) h = 3 cm ⇒ t = 3arccos (3/7) / π =  1.08s← answer


3) h = 5 cm ⇒ 3arccos (5/7) / π = 0.74 s← answer



c. Find the times at which the weight is at a height of 1 cm, of 3 cm, and of 5 cm below the rest position for the second time.

Use the periodicity property of the function.

The periodicity of cos(π/3 t) is 6.


So, the second times are:


1) h = 1 cm, t = 6 + 0.45 s = 6.45 s ← answer


2) h = 3 cm ⇒ 6 + 1.08 s = 7.08 s← answer


3) h = 5 cm ⇒ t = 6 + 0.74 s = 6.74 s ← answer