The height in metres of the water in a harbour is given approximately by the formula d=6+3cos30t, where t is the time in hours from. Find
a) The height if the water at 9:45pm
b) The highest and the lowest water level and occur
"where t is the time in hours from....." is incomplete
Did you mean from noon ?
Let's assume it is after noon, else make the necessary adjustments
at 9:45 pm , t = 9.75
set your calculator to radians
d = 3.163 metres
The min of 3cos 30t is -3 and its max is +3
So the max height is 9 m and the lowest is 3
Can you please explain to me why the maximum value of d occurs when the value of the cosine function is 1?
Thank you
a) Ah, 9:45pm, the perfect time for some high tide comedy! Anyway, to find the height of the water at 9:45pm, we just need to substitute t = 9.75 (since it's 9 hours and 45 minutes past the starting time) into the formula. So, d = 6 + 3cos(30 * 9.75). Crunching the numbers with some mathematical magic, we get d ≈ 5.988 meters. So, approximately 5.988 meters is the height of the water at 9:45pm. Let's hope it's not high enough to turn the whole harbor into a swimming pool!
b) Now, to find the highest and lowest water levels, we need to remember that the cosine function has a maximum value of 1 and a minimum value of -1. Since the formula is d = 6 + 3cos30t, the highest water level occurs when cos30t equals 1 (maximum value), and the lowest water level happens when cos30t equals -1 (minimum value).
Setting cos30t = 1, we have 1 = 1, which means that the maximum height is 6 + 3(1) = 9 meters. Time for some high tide fun!
Next, setting cos30t = -1, we have -1 = -1. This means that the minimum height is 6 + 3(-1) = 3 meters. So, it looks like the water level can go from 3 meters below sea level to 9 meters above sea level. Quite the rollercoaster ride, isn't it? Just make sure to keep your balance and enjoy the show!
To find the height of the water at a specific time and to determine the highest and lowest water levels, we can use the given formula d=6+3cos30t.
a) The height of the water at 9:45 pm:
First, we need to convert the time to hours. Since there are 60 minutes in an hour, 45 minutes is equal to 45/60 = 0.75 hours. Therefore, the time in hours is 9 + 0.75 = 9.75 hours.
Now, we can substitute the value of t into the formula: d = 6 + 3cos(30 * 9.75).
To find the value of cos(30 * 9.75), we need to convert 30 * 9.75 to radians since the cosine function expects the input in radians.
One full rotation is equal to 2π radians, and there are 360 degrees in a full rotation. Therefore, 30 degrees is equal to 30 * (2π/360) = π/6 radians.
Now, we can substitute the value of π/6 into the formula: d = 6 + 3cos(π/6 * 9.75).
Using a calculator or a mathematical software, we can find the value of cos(π/6 * 9.75) ≈ 0.130526, rounded to six decimal places.
Therefore, the height of the water at 9:45 pm is d = 6 + 3 * 0.130526 ≈ 6.391578 meters.
b) The highest and lowest water levels:
The given formula for the height of the water, d = 6 + 3cos30t, tells us that the highest water level occurs when the cosine function has a maximum value of 1, and the lowest water level occurs when the cosine function has a minimum value of -1.
The cosine function has a maximum value of 1 at 0 degrees/0 radians, 360 degrees/2π radians, etc.
To find when the cosine function has a maximum value, we need to solve the equation 30t = 0.
Dividing both sides by 30, we get t = 0.
Therefore, the highest water level occurs at t = 0 hours.
Similarly, the cosine function has a minimum value of -1 at 180 degrees/π radians, 540 degrees/3π radians, etc.
To find when the cosine function has a minimum value, we need to solve the equation 30t = π.
Dividing both sides by 30, we get t = π/30.
Therefore, the lowest water level occurs at t = π/30 hours.
As t is measured in hours, you can leave the values as t = 0 and t = π/30, respectively.
Hence, the highest water level occurs at t = 0 hours, and the lowest water level occurs at t = π/30 hours.