The vertical distance from the dock to the boat's mast reaches its highest value of $-27 \mathrm{~cm}$ every $3$ seconds. The first time it reaches its highest point is after $1$ . $3$ seconds. Its lowest value is $-44 \mathrm{~cm}$ . $\newline$ Find the formula of the trigonometric function that models the vertical height $H$ between the dock and the boat's mast $t$ seconds after Antonio starts his stopwatch. Define the function using radians. $\newline$ $H(t)=\square$ $\newline$ What is the vertical distance $2$ . $5$ seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places.

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Q. The vertical distance from the dock to the boat's mast reaches its highest value of

-27 \mathrm{~cm}

every

3

seconds. The first time it reaches its highest point is after

1

3

seconds. Its lowest value is

-44 \mathrm{~cm}

\newline

Find the formula of the trigonometric function that models the vertical height

H

between the dock and the boat's mast

t

seconds after Antonio starts his stopwatch. Define the function using radians.

\newline

H(t)=\square

\newline

What is the vertical distance

2

5

seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places.

Amplitude Calculation: The question_prompt: What is the vertical distance from the dock to the boat's mast $2.5$ seconds after Antonio starts his stopwatch?
Period and Angular Frequency: First, we need to determine the amplitude of the trigonometric function. The amplitude is half the distance between the highest and lowest values of the function. $\newline$ Amplitude $A$ = $(\text{Highest value} - \text{Lowest value}) / 2$ $\newline$ $A = (-27\,\text{cm} - (-44\,\text{cm})) / 2$ $\newline$ $A = (44\,\text{cm} - 27\,\text{cm}) / 2$ $\newline$ $A = 17\,\text{cm} / 2$ $\newline$ $A = 8.5\,\text{cm}$
Phase Shift Calculation: Next, we find the period of the function. The period $T$ is the time it takes for the function to repeat its values, which is given as every $3$ seconds. $\newline$ $T = 3$ seconds $\newline$ To find the angular frequency $\omega$ , we use the formula $\omega = 2\pi / T$ . $\newline$ $\omega = 2\pi / 3$
Negative Vertical Distance Adjustment: The vertical distance reaches its highest value at $1.3$ seconds for the first time. This means the phase shift $(\varphi)$ is to the left by $1.3$ seconds. Since the cosine function starts at its maximum value, we will use the cosine function for our model. $\newline$ $\varphi = -1.3$ seconds $\newline$ To convert the phase shift into radians, we multiply by the angular frequency. $\newline$ $\varphi \text{ (in radians)} = \omega \times \varphi \text{ (in seconds)}$ $\newline$ $\varphi \text{ (in radians)} = (2\pi / 3) \times (-1.3)$ $\newline$ $\varphi \text{ (in radians)} = -2\pi / 3 \times 1.3$
Trigonometric Function Formulation: The vertical distance is negative because it is measured from the dock downwards. Therefore, we need to include a negative sign in our amplitude to reflect this direction. $\newline$ The general form of the trigonometric function is: $\newline$ $H(t) = A \cdot \cos(\omega t + \varphi) + D$ $\newline$ where $D$ is the vertical shift, which is the average of the highest and lowest values. $\newline$ $D = (\text{Highest value} + \text{Lowest value}) / 2$ $\newline$ $D = (-27\,\text{cm} + (-44\,\text{cm})) / 2$ $\newline$ $D = (-71\,\text{cm}) / 2$ $\newline$ $D = -35.5\,\text{cm}$
Vertical Distance Calculation: Now we can write the function that models the vertical height $H$ between the dock and the boat's mast $t$ seconds after Antonio starts his stopwatch. $H(t) = -8.5 \cdot \cos\left(\frac{2\pi}{3}t - \frac{2\pi}{3} \cdot 1.3\right) - 35.5$
Cosine Value Calculation: To find the vertical distance $2.5$ seconds after Antonio starts his stopwatch, we substitute $t = 2.5$ into the function. $\newline$ $H(2.5) = -8.5 \times \cos((2\pi / 3)(2.5) - 2\pi / 3 \times 1.3) - 35.5$ $\newline$ $H(2.5) = -8.5 \times \cos((5\pi / 3) - (2.6\pi / 3)) - 35.5$ $\newline$ $H(2.5) = -8.5 \times \cos(2.4\pi / 3) - 35.5$
Final Vertical Distance: We calculate the cosine value and the final height. $\newline$ $H(2.5) = -8.5 \times \cos(0.8\pi) - 35.5$ $\newline$ $H(2.5) = -8.5 \times (-0.5877852523) - 35.5$ (using a calculator for $\cos(0.8\pi)$ ) $\newline$ $H(2.5) = 4.996679642 - 35.5$ $\newline$ $H(2.5) = -30.50332036$ cm $\newline$ Rounded to two decimal places, the vertical distance is $-30.50$ cm.