A)

The ocean tides are due to the gravitational force exerted by the Moon on the water of the oceans. When the Moon is facing the ocean, the water is attracted towards the Moon, and therefore the sea level rises, causing high tide.

The tide follows a periodic pattern because the Earth rotates around its axis, therefore, there are times in which the water is facing the Moon (so, there is high tide), and times during which is not facing the Moon (so, there is low tide).

C)

In this problem:

The lowest low tide was first measured on July 11, at -1.4 feet.

The highest low tide was first measured on July 4, at 1.8 feet.

If we take July 1 as "day 0", then we should choose a sine function. This is because at July 4, the tide is at its maximum, while at July 11, the tide is at its minimum; this means that the tide is at "zero" value 3.5 days after and before every maximum/minimum.

So, the tide has value zero 3.5 days before July 4, which means on July 1. Since the sine function is zero for t = 0, this means we should choose a sine.

D)

The amplitude of a periodic function is the maximum displacement reached by the function relative to its equilibrium position.

Here, we have:

- Average value of the function: 0.2 feet

- Maximum value: 1.8 feet

- Minimum value: -1.4 feet

This means that the maximum displacement of the function (and so, the amplitude) is:

(1.6 feet)

Here it means that the maximum tide is 1.6 feet higher than the average value (0.2 ft) and the minimum tide is 1.6 feet lower than the average value (0.2 ft).

E)

The period of a periodic function is the time the function takes to complete one oscillation.

The period of a periodic function is also equivalent to twice the time the function takes to go from its maximum value to its minimum value.

In this problem:

- The maximum tide occurs on July 4

- The minimum tide occurs on July 11

So, the time between minimum and maximum is 7 days

Therefore, the period is twice this value:

Here it means the tide has a maximum (or a minimum) every 14 days.

F)

The vertical shift of the function is the displacement of the average value from the position zero.

Here, we are told that the average of the tide during July is 0.2 feet.

This means that the vertical shift of the function is +0.2 feet (this represents the "zero value" of the sine function).

Here it means that when there is no tide, the value of the sea level is +0.2 feet

G)

The phase shift of a periodic function corresponds to the horizontal shift of the function, i.e. the angle at which the function has its zero value.

In this problem, we notice that the function should be zero at t = 0.5 (3.5 days before the first maximum, which occurs at t = 4 days); therefore, the phase shift in this case is -0.5 days.

Here it means that the first zero of the periodic function (the sine) occurs after 0.5 days; so, the tide is at its average height (0.2 ft) at t = +0.5 days.

H)

The sine function can be written as

where

A = 1.6 ft is the amplitude

is the period

is the vertical shift

is the phase shift

t is the time (in days)

Substituting all the values found so far, we find the expression for the function:

I)

See graph in attachment.

J)

Here we said that the period of the tide function is 14 days: this means that the time between two consecutive maxima of the tide is 14 days.

During this time, however, the tide crosses 2 times the zero-level (which is 0.2 feet) of the tide. This means that the tide is at the average height (0.2 feet) every 7 days.

So, starting from 0, the tide is at its average height on the following days:

t = 0.5

t = 7.5

t = 14.5

t = 21.5

t = 28.5

The decimal value means "in the middle of the day".

K)

There are various reasons for which the height recorded vary from the function created.

For example, the tides do not depend only on the force of gravity exerted by the Moon, but also on the force of gravity exerted by the Sun. For instance, when the Sun and the Moon are along the same line, the force of gravity is stronger and the high tide is even higher; vice-versa, when the Sun and the Moon are perpendicular to each other (relative to the Earth), the force of gravity is less and the tides are lower than their maximum/minimum values.