(10 points total)\na boat docked on a pier ri...
(10 points total)\na boat docked on a pier rises and falls as the tide rises and falls. when the tide is high, the base of the boat measures 22 feet above the ocean floor. six hours later, when the tide is low, the base of the boat measures 8 feet above the ocean floor. the sinusoidal function f models the distance between the base of the boat and the ocean floor as a function of t in hours. assume the first high tide is at time t = 0.\n\npart a:\n• identify the midline and amplitude for the scenario. (2 points)\n• identify the period and any phase shift for the scenario. (1 point)
Answer
# Explanation:
## Step1: Find the mid - line
The mid - line is the average of the maximum and minimum values. The maximum value (high tide) is 22 feet and the minimum value (low tide) is 8 feet. So, the mid - line $y=\frac{22 + 8}{2}$.
$y=\frac{30}{2}=15$
## Step2: Find the amplitude
The amplitude is the distance from the mid - line to the maximum or minimum value. So, $A=\frac{22-8}{2}$.
$A=\frac{14}{2}=7$
## Step3: Find the period
The time from high tide to low tide is 6 hours. Since a full cycle (from high tide to next high tide) is twice the time from high tide to low tide, the period $T = 12$ hours.
## Step4: Find the phase shift
Since the first high tide is at $t = 0$, there is no phase shift, so the phase shift is 0.
# Answer:
Mid - line: $y = 15$; Amplitude: $A=7$; Period: $T = 12$ hours; Phase shift: 0