Question

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?

Explanation:

Step1: Analyze the function form

The function is \( h(t) = 5\cos(0.5t - 2)+7 \), which is a cosine function of the form \( y = A\cos(Bt - C)+D \). Here, \( A = 5 \) (amplitude), \( B = 0.5 \) (affects period), \( C = 2 \) (phase shift), and \( D = 7 \) (vertical shift).

Step2: Determine key features

• Vertical shift: The midline is \( y = 7 \) (since \( D = 7 \)).
• Amplitude: The maximum height is \( 7 + 5=12 \) and minimum is \( 7 - 5 = 2 \).
• Period: The period of \( \cos(Bt) \) is \( \frac{2\pi}{B} \), so here \( \frac{2\pi}{0.5}=4\pi\approx12.57 \) hours.
• Phase shift: The phase shift is \( \frac{C}{B}=\frac{2}{0.5} = 4 \) hours (shift to the right).

We need to check which graph has a midline at \( y = 7 \), amplitude 5, period \( 4\pi \), and phase shift 4.

Answer:

(Assuming options are graphs, we would identify the graph with midline \( y = 7 \), max 12, min 2, period ~12.57, and shifted right by 4. Since no graphs are shown here, but in a typical problem, we'd pick the graph matching these features. For example, if one graph has midline 7, oscillates between 2 and 12, period ~12.5, and starts its cycle shifted right, that's the one.)