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The motion of a ship riding at anchor can be modeled by y=25-4 cos (pi/6)t, where y is the water depth in feet and t is the time in hours. Consider a day in which t=0 represents 12:00 midnight. At what time during that day will the water under the ship be the deepest and what time will it be shallowest? What will the water’s depth be at these times?

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tramserran

Answer: 29 feet (deepest) at 6 am

              21 feet (shallowest) at Midnight & Noon

Step-by-step explanation:

y = -4 cos (π/6)t +25

Amplitude (A) = 4    

-A means it is a reflection over the x-axis (starts at minimum)

Period = 2π/B   →  Period = 12

Phase shift = C/B   →  Phase Shift = 0

Midline (D) = 25

Midline (D) ± Amplitude (A) = Max & Min

Max: 25 + 4 = 29

Min: 25 - 4 = 21

Change the coordinates of y = cos (x) as follows:

  • x-value: Add C then divide by B
  • y-value: Multiply by A then add D

Note that the equation shows: A = -4, B = π/6, C = 0, D = 25

[tex]\begin{array}{c|ccc|cl}\underline{\quad x\quad}&\underline{\quad y\quad}&&\underline{\quad (x+C)/B\quad}&\underline{\quad Ay+D\quad}\\0&1&&0&21&minimum\\\pi&0&&3&25&midline\\\pi &-1&&6&29&maximum\\3\pi/2&0&&9&25&midline\\2\pi&1&&12&21&minimum\\\end{array}[/tex]

Answer:

Shallowest:midnight+ noon

height:21 feet

Deepest: 6am+ 6pm

height: 29 feet

Step-by-step explanation:

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