The figure shows the depth of water at the end of a boat doc

The figure shows the depth of water at the end of a boat dock. The depth is 12 feet at low tide and 22 feet at high tide. On a certain day, low tide occurs at 6 AM and high tide at noon. If y represents the depth of the water x hours after midnight, use a cosine function of the form y = A cos Bx + D to model the water\'s depth. Write the equation for this problem in the form y = A cos Bx + D y = (Type an expression using x as the variable. Type an exact answer using x as needed. Use integers or fractions for any numbers in the expression.)

Solution

y = AcosBx +D

amplitude, A = ( Yhigh tide - Y low tide)/2 = (22-12)/2 = 5 feet

Midline , D = (22 +12)/2 =17

So, High tide = 5 +17 and low tide = 17 -5 = 12 feet

So, amplitude A and D satisfy the values of graph.

Half Time Period = 12 - 6 = 6 hrs

So, Period = 2*6 = 12 hrs

So, B = 2pi/period = 2pi/12 = pi/6

So, y = 5cos(pi*x/6) + 17