(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)\n\npart b:\n • identify possible coordinates for a, b, c, d, and e. (2 points)\n • find the values of the constants a, b, c, and d for the function f(t)=a cos(b(t + c))+d. (2 points)
Answer
# Explanation:
## Step1: Calculate 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 $y=\frac{22 + 8}{2}=\frac{30}{2}=15$ feet.
## Step2: Calculate amplitude
The amplitude is the distance from the mid - line to the maximum or minimum value. So, $A=\frac{22-8}{2}=\frac{14}{2}=7$ feet.
## Step3: Determine period
The time from high tide to low tide is 6 hours. Since the time from high tide to low tide is half of the period, the period $T = 12$ hours. And since the first high tide is at $t = 0$, there is no phase shift, $\varphi=0$.
## Step4: Find coordinates
Assuming the function is $y = a\cos(b(t + c))+d$, with $t$ in hours. For point $A$ (high tide at $t = 0$), the coordinate is $(0,22)$. For point $B$ (mid - line value halfway between high and low tide), $t=3$ and $y = 15$, so the coordinate is $(3,15)$. For point $C$ (low tide), $t = 6$ and $y=8$, so the coordinate is $(6,8)$. For point $D$ (mid - line value on the rise from low tide), $t = 9$ and $y = 15$, so the coordinate is $(9,15)$. For point $E$ (next high tide), $t=12$ and $y = 22$, so the coordinate is $(12,22)$.
## Step5: Find function constants
The general form of the cosine function is $y=a\cos(b(t + c))+d$. Amplitude $a = 7$, period $T=\frac{2\pi}{b}$, since $T = 12$, then $b=\frac{2\pi}{12}=\frac{\pi}{6}$. There is no phase shift, so $c = 0$. The mid - line is $y = 15$, so $d = 15$.
# Answer:
Part A:
- Mid - line: $y = 15$ feet, Amplitude: $7$ feet, Period: $12$ hours, Phase shift: $0$
Part B:
- $A=(0,22)$, $B=(3,15)$, $C=(6,8)$, $D=(9,15)$, $E=(12,22)$
- $a = 7$, $b=\frac{\pi}{6}$, $c = 0$, $d = 15$