After spending the day building sand castles,
Oceana wants to take an evening walk with
a friend along the shoreline.
Oceana knows that one stretch along the
shore is quite rocky. At that point, the rocks
jut into the ocean. To walk around them,
a person has to follow a path that is 14 feet
below the average waterline.
If Oceana and her friend don’t want to get
wet, they need to take their walk when the
waterline is 14 feet or more below the average
waterline. What is the time period during
which they can take their walk?
Remember that the position of the waterline
over the course of the day is given by the
equation w(t) 5 20 sin (29t), where the
distance is measured in feet and t represents
the number of hours elapsed since midnight.

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oeerivona oeerivona · Answer 1 · 2022-02-07T19:25:07+01:00

The function for the water level is a periodic function, such

that the water is at a level on more than one occasion.

Correct response:

The time period during which they can take their walk is between 4:00 p.m. and 5.09 p.m. in the evening.

The path below the average water line a person must follow to go round the rock = 14 feet

The function for the level of water is; w(t) = 20·sin(29·t)

Where;

t = The number of hours that have elapsed after midnight

When w(t) = 14, we have;

14 = 20·sin(29·t)

[tex]sin(29 \cdot t) = \dfrac{14}{20} = \mathbf{0.7}[/tex]

29·t = arcsine(0.7) ≈ 44.427

[tex]t = \mathbf{\dfrac{44.427}{29}} \approx 1.532[/tex]

Solving using an online application, we have;

[tex]t \approx \mathbf{1.53 + \dfrac{360}{29} \cdot n}, \ 4.675 + \dfrac{360}{29} \cdot n[/tex]

By using the above equation, and from the graph of the

function; w(t) = 20·sin(29·t), we have;

The time period during which they can take their walk are;

When n = 0

Between t = 1.53 hours (1.53 a.m.), and t = 4.675 hours after midnight (before daybreak)

When n = 1

[tex]t \approx 1.53 + \dfrac{360}{29} \approx 13.944 , \ 4.675 + \dfrac{360}{29} \approx 17.09[/tex]

Between 13.944 hours (approximately 2:00 p.m.) and 17.09 hours (approximately 5:09 p.m.)

Therefore;

The time in the evening at which they can take their walk is before 5:09 p.m. which is between 4:00 p.m. and 5.09 p.m.

Learn more about sinusoidal functions here;

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