The equation models the height h in centimeters after t seconds of a weight attached to the end of a spring that has been stretc
hed 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.
This is the missing equation that models the hieght and is misssing in the question:
<span>h= 7cos(π/3 t) </span>
Answers:
<span>a. Solve the equation for t. </span>
<span>1) Start: h= 7cos(π/3 t) </span>
2) Divide by 7: (h/7) = <span>cos(π/3 t) </span>
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.
<span>1) h = 1 cm ⇒ t = 3 arccos(1/7) / π</span>
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 <span>cos(π/3 t) is 6. </span><span> </span><span> </span><span>So, the second times are: </span><span> </span><span> </span><span>1) h = 1 cm, t = 6 + 0.45 s = 6.45 s← answer </span>
The lengths of the second triangle is just half the length of the first triangle, from the center. As it is just resizing, the angles would not change, resulting in the same triangle, except for size.
Since there are 365 possible slots for people to have their birthday on, the worst case happens when all 365 people have different birthday. This means the 366th person would have their birthday falls on any of other’s birthday. Hence, kk must be at least 366.
