The depth of the ocean at a swim buoy can be modelled by the function 𝑑(𝑡) = 2 sin ((𝜋/12)𝑡) + 3 , where d is the depth of water in metres and t is the time in hours, t∈[0,24]. If t = 0 corresponds midnight, what is the depth
of water at 2 pm?
just plug in t=14
To find the depth of water at 2 pm, we need to substitute the value of t = 14 into the function 𝑑(𝑡) = 2sin((𝜋/12)𝑡) + 3.
Step 1: Convert the time to hours since midnight
Since t = 0 corresponds to midnight, we need to convert 2 pm to hours since midnight. To do this, we add 12 to the number of hours in 2 pm:
2 pm + 12 hours = 14 hours
Step 2: Substitute the value of t into the function
Now, substitute t = 14 into the function:
𝑑(14) = 2sin((𝜋/12) * 14) + 3
Step 3: Evaluate the expression
Use a scientific calculator to calculate sin((𝜋/12) * 14), then multiply it by 2 and add 3.
The final value you get is the depth of water in meters at 2 pm.
To find the depth of the water at 2 pm, we need to substitute t = 14 into the equation 𝑑(𝑡) = 2 sin ((𝜋/12)𝑡) + 3.
Let's calculate this step-by-step:
1. Convert 2 pm to the 24-hour format: 2 pm is 14 in the 24-hour format.
2. Substitute t = 14 into the equation: 𝑑(14) = 2 sin ((𝜋/12) * 14) + 3.
3. Simplify the equation: 𝑑(14) = 2 sin (𝜋 * (14/12)) + 3.
𝑑(14) = 2 sin (𝜋 * (7/6)) + 3.
4. Evaluate sin (𝜋 * (7/6)): To do this, remember that sin (𝜋/6) = 1/2, and sin (𝜋/3) = √3/2.
Since (7/6) is between (𝜋/6) and (𝜋/3), we can approximate it as √3/2.
𝑑(14) = 2 * (√3/2) + 3.
𝑑(14) = √3 + 3.
5. Calculate the value: 𝑑(14) = √3 + 3 ≈ 4.732.
Therefore, the depth of water at 2 pm is approximately 4.732 meters.