throughout the day, the depth of water at the end of a dock varies with the tides. the function…

Answer:

We need to analyze the key - features of the function (h(t)=5\cos(0.5t - 2)+7) to determine the correct graph.

The general form of a cosine function is (y = A\cos(Bx - C)+D).

For the function (h(t)=5\cos(0.5t - 2)+7):

• Amplitude (A = 5). This means the maximum deviation from the mid - line is 5.
• The mid - line is (y = D=7). The maximum value of the function is (y_{max}=A + D=5 + 7 = 12) and the minimum value is (y_{min}=-A + D=-5 + 7 = 2).

We can also find the period of the function. The period of a cosine function (y=\cos(Bx - C)+D) is given by (T=\frac{2\pi}{B}). Here, (B = 0.5), so (T=\frac{2\pi}{0.5}=4\pi\approx12.57).

Now, let's check the graph: The graph should oscillate between (y = 2) and (y = 12) with a mid - line at (y = 7) and a period of approximately (12.57) hours.

If the given graph has these characteristics (oscillating between 2 and 12 with mid - line at 7 and a period close to 12.57), then it is the correct graph. Without seeing all the options, we can't definitively say it is the correct one, but we can analyze it based on the above - mentioned properties.

Explanation:

Step1: Identify the amplitude

The amplitude (A) of (y = A\cos(Bx - C)+D) is (|A|). Here (A = 5), so the function oscillates 5 units above and below the mid - line.

Step2: Identify the mid - line

The mid - line of the function (y = A\cos(Bx - C)+D) is (y = D). Here (D = 7), so the mid - line is (y = 7).

Step3: Calculate the period

Using the formula (T=\frac{2\pi}{B}), with (B = 0.5), we get (T=\frac{2\pi}{0.5}=4\pi\approx12.57). This is the time it takes for the function to complete one full cycle.