Question

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?

Answer 1

Shallowest:midnight+ noon

height:21 feet

Deepest: 6am+ 6pm

height: 29 feet

Explanation:

Answer 2

Answer: 29 feet (deepest) at 6 am

21 feet (shallowest) at Midnight & Noon

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

`\begin{array}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}`