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

throughout the day, the depth of water at the end of a dock varies with the tides. the function $h(t)=5cos(0.5t - 2)+7$ represents the height, in feet, of the water $t$ hours after midnight. which graph shows the height of the water at the dock at any time after midnight?

throughout the day, the depth of water at the end of a dock varies with the tides. the function $h(t)=5cos(0.5t - 2)+7$ represents the height, in feet, of the water $t$ hours after midnight. which graph shows the height of the water at the dock at any time after midnight?

Answer

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}=7 + 5=12) and the minimum value is (y_{min}=7 - 5 = 2).

We can also find the period of the function. The period of a cosine function (y = A\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:

  1. The mid - line of the graph should be at (y = 7).
  2. The maximum value of the function should be 12 and the minimum should be 2.
  3. The period is approximately 12.57.

Based on these features, we can match the function with the correct graph among the given options (not all options are shown in the problem statement, but we can analyze the provided graph). The provided graph has a mid - line around (y = 7), maximum value around 12 and minimum value around 2, and a period that seems to be around 12 hours which is close to our calculated period of approximately 12.57 hours.

Since no other graphs are provided, if this is the only graph available, it is likely the correct one as it satisfies the key features of the function (h(t)=5\cos(0.5t - 2)+7).

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. [A = 5]

Step2: Identify the mid - line

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

Step3: Calculate the period

The period (T) of (y = A\cos(Bx - C)+D) is (T=\frac{2\pi}{B}). Given (B = 0.5), we have (T=\frac{2\pi}{0.5}=4\pi\approx12.57). [T=\frac{2\pi}{0.5}]