The depth of the water in a fishing port is usually 28 meters. The tidal movement is represented by 𝑆 equals four times sin of 15𝑛 degrees plus 28, where 𝑛 is the time elapsed in hours after midnight. How many times a day is the depth of the water 24 meters?
We’re given a function that models the movement of the water. And that is, 𝑆 equals four times sin of 15𝑛 degrees plus 28. It would be helpful to first establish the parameters for 𝑛, what 𝑛 can be. We’re wondering how many times a day the depth of the water is 24 meters. And so, we’re dealing with one day, 24 hours. And if our 𝑛 is the time elapsed in hours, then we can have an 𝑛 variable that is greater than or equal to zero or less than or equal to 24. We can use this information to find out what 15𝑛 degrees must then be. 15 times zero would be zero degrees. 15 times 24 is 360 degrees. And so, we can say that we’re dealing with sine functions of one rotation, from zero degrees to 360 degrees.
If we think about sketching a sine function, we take a range of 𝑥 values from zero to 360 degrees. Sin of zero degrees is zero. We know that sin of 90 degrees is one. Sin of 180 degrees is zero. Sin of 270 degrees is negative one. And sin of 360 degrees is also zero. This is the graph of 𝑦 equals the sin of 𝑥 degrees. If we wanted 𝑦 equals four times the sin of 𝑥 degrees, well sin of zero degrees times four is still zero. Sin of 90 degrees is one, and one times four is four. Four times the sin of 180 degrees is still zero, as would be four times the sin of 360 degrees. Four times the sin of 270 degrees would be equal to negative four. It would be four times negative one.
So, we see the way multiplying four by the sine function stretches it in the vertical direction. And now, we need to deal with this 28 meters. We know that the water is regularly at 28 meters. When the water starts at 28 meters and makes an increase of four meters, it’s at 32 meters. And when the water makes a decrease of four meters, it’s set at 24 meters. Based on this model, the tide only goes down to 24 meters one time in a 24-hour period. The question was not asking what time of day the tide hits 24 meters, but how many times a day. And based on this information, it only happens one time.