Must have a graph, and write an equation

Question

the time interval between high tide and low tide in a river harbor is approximately 7 hours. The high tide depth of 16 feet occurs at noon and the average harbor depth is 11 feet. Write an equation modeling this relationship. What time will the river harbor be at low tide? 

Answer 1

This problem requires you to write an equation for a sine wave. Sine waves model periodic events (events that repeat), like tides. You'll need to find the vertical and horizontal shifts and stretches that can transform the basic sine equation y = sinx into one that models your situation.

 

The basic sine equation has its middle (average) at 0, starts at its average point then proceeds to its maximum, has a period of 2π (which means it takes 2π to complete one cycle), and has an amplitude of 1 (which means its maximum and minimum are 1 unit above and below the average, at 1 and -1).

 

 

Let's start with vertical shift and vertical stretch (amplitude).

Here, your sine wave's middle or average value is 11 ("the average harbor depth is 11 feet"). That's a vertical shift up of 11.

Since the sine wave reaches 16 - 11 = 5 feet above its average, its amplitude is 5. That's a vertical stretch of 5.


Now, let's work on the sine wave's horizontal shift and horizontal stretch (period).

Normally, the sine wave is at its middle point at x = 0, from which it increases to its maximum. Here, the problem doesn't mention anything about calling some time "time 0," so we don't have to shift it at all. We can just choose to start the sine wave at 0. There is no horizontal shift.

Since there are 7 hours between high and low tides, it takes 14 hours to get from high tide to the next high tide. That's how long it will take to get from one maximum to another. The stretch coefficient for the period, b, is:

(2π)/b = period
(2π)/b = 14
2π = 14b
(2π)/14 = b

π/7 = b

That's a horizontal stretch of π/7.


Let's take all that information and put it into the sine's equation. A normal sine wave has the equation:

y = sin(x)

A transformed sine wave has the equation:

y = asin(bx + c) + d

where a = vertical stretch (amplitude), b = horizontal stretch (period), c = horizontal shift (positive is left, negative is right), and d = vertical shift (positive is up, negative is down).

For us, a = 5, b = π/7, c = 0 (no shift), and d = 11 (shift up). Plug those numbers in to get your equation:

y = 5sin((π/7)x) + 11

 

 

As for the second part, the time the river will be at low tide, you don't really need the equation for that. Since it takes 7 hours to get from high tide to low tide and high tide occurs at noon, low tide will occur 7 hours after noon, at 7 PM.