During a storm a ship is being tossed up and down by huge waves. The vertical motion of the ship is given by the equation: h = 21cos((2π/5)t) where h is height above sea level in feet at time t minutes.
1) Give the first 2 times that the ship is at sea level.
2) Give a general expression for the times at which the ship is at sea level.
3) When does the ship first fall to 17 feet below sea level?
4) Is the ship above sea level or below sea level at time 12 minutes?
(1) well, you want 2π/5 t = π/2 and 3π/2
so, that makes t = 5/4 for the first time.
Since the period is 2π/(2π/5) = 5, the second time is 5/2 later, at t=15/4
(2) from the period, this occurs at 5/4 + 5/2 k for integer k
(3) just solve for t in
21cos((2π/5)t) = -17
(4) same for t=2. h positive or negative?
See the graph at
http://www.wolframalpha.com/input/?i=21cos%28%282%CF%80%2F5%29t%29+for+0%3C%3Dt%3C%3D15
1) To find the first two times the ship is at sea level, we need to find the values of t for which h = 0. We can set the equation h = 21cos((2π/5)t) equal to zero and solve for t.
21cos((2π/5)t) = 0
For cosine to be zero, the angle inside the function must be either π/2 or 3π/2.
So we have two possible equations:
(2π/5)t = π/2
Solving for t: t = (π/2) * (5/(2π)) = 5/4 minutes
(2π/5)t = 3π/2
Solving for t: t = (3π/2) * (5/(2π)) = 15/4 minutes
Therefore, the first two times the ship is at sea level are t = 5/4 minutes and t = 15/4 minutes.
2) To find a general expression for the times at which the ship is at sea level, we need to consider that the cosine function has a period of 2π. In our case, the period is (2π/5).
To find the general expression, we can start with the first two times we found (t = 5/4 and t = 15/4 minutes) and add the period to each subsequent time:
General expression: t = (5/4) + (2π/5)n, where n is an integer representing the number of periods.
This expression represents all the times when the ship is at sea level.
3) To find when the ship falls 17 feet below sea level, we need to find the value of t for which h = -17. We can set the equation h = 21cos((2π/5)t) equal to -17 and solve for t.
21cos((2π/5)t) = -17
Dividing both sides by 21, we get:
cos((2π/5)t) = -17/21
Using the inverse cosine function, we can find the angle for which the cosine is equal to -17/21:
(2π/5)t = arccos(-17/21)
Solving for t, we get:
t = (arccos(-17/21))/(2π/5)
Calculating this expression will give us the time when the ship first falls 17 feet below sea level.
4) To determine if the ship is above or below sea level at time 12 minutes, we can substitute t = 12 into the equation for h:
h = 21cos((2π/5)t)
h = 21cos((2π/5)*12)
Calculating this expression will give us the height of the ship at time 12 minutes. If the value is greater than zero, the ship is above sea level; if it's less than zero, the ship is below sea level.