Question

The water level f, in feet, at a certain beach is modeled by the function


`f(t)=2sin(2\pi t)/24`, where is the number of hours since the level was measured.

What is the amplitude of the function? What does it mean in this context?
What is the period of the function? What does it mean in this context?
Sketch a graph of the function over 72 hours.

Answer 1

1. Amplitude = 2 feet (max height level of water on the beach)

2. Periodicity = 24 hours (water level reaches 2 ft max every 24 hours or 0 feet minimum every 24 hours

3. Graph attached with explanation

Explanation:

For a sine function, sin(x), the maximum value is 1 when x is π/2, 5π/2, 9π/2...

The minimum value is -1 at x = 0, π, 3π/2, 2π,...

a sin(x) will have the maximum value(amplitude) of |a · 1| = |a|

The given function is

`f(t) = 2 \sin\left((2\pi t)/(24)\right)\\\\(2\pi t)/(24) = (\pi t)/(12)\\\\\textrm{So the function is:}}\\\\f(t) = 2 \sin \left((\pi t)/(12)\right)`

1. The amplitude is 2 and this is the maximum water level in feet at the beach. This occurs when
`\textstyle {(\pi t )/(12) = (\pi)/(2)}`

or, when t = 6. It repeats every 24 hours (see #2)

2. The periodicity of a function is when the function starts repeating itself

The periodicity of sin(x) is 2π i.e. the sinusoidal wave starts repeating itself every 2π radians

Since our sine function has πt/12 as its argument we have that
πt/12 = 2π

or t = 24

So every 24 hours, the water level which starts at 0 feet at 0 hours, becomes 0 again

3) Graph of the function attached. Note that water level cannot be negative which is the sine value from π/2 to 2π so we have to set it equal to 0

Water level goes to zero every 12 hours and stays at zero for 12 hours when it starts rising again.

As can be seen from the graph, every 24 hours, the water level reaches a peak height of 2 feet

I hope that this helps