The vertical distance from the dock to the boat's mast reaches its highest value of -27cm every 3 seconds. The first time it reaches its highest point is after 1.3 seconds. Its lowest value is -44cm. 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. H(t)= ◻ What is the vertical distance 2.5 seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places.

Question

The vertical distance from the dock to the boat's mast reaches its highest value of  -27cm every 3 seconds. The first time it reaches its highest point is after 1.3 seconds. Its lowest value is  -44cm. 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.  H(t)=  ◻ 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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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. Amplitude (A) = (Highest value - Lowest value) / 2 A = (-27cm - (-44cm)) / 2 A = (44cm - 27cm) / 2 A = 17cm / 2 A = 8.5cmPhase 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. T = 3 seconds To find the angular frequency (ω), we use the formula ω = 2π / T. ω = 2π / 3Negative Vertical Distance Adjustment: The vertical distance reaches its highest value at 1.3 seconds for the first time. This means the phase shift (φ) 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. φ = -1.3 seconds To convert the phase shift into radians, we multiply by the angular frequency. φ (in radians) = ω * φ (in seconds) φ (in radians) = (2π / 3) * (-1.3) φ (in radians) = -2π / 3 * 1.3Trigonometric 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. The general form of the trigonometric function is: H(t) = A * cos(ωt + φ) + D where D is the vertical shift, which is the average of the highest and lowest values. D = (Highest value + Lowest value) / 2 D = (-27cm + (-44cm)) / 2 D = (-71cm) / 2 D = -35.5cmVertical 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 * cos((2π / 3)t - 2π / 3 * 1.3) - 35.5Cosine Value Calculation: To find the vertical distance 2.5 seconds after Antonio starts his stopwatch, we substitute t = 2.5 into the function. H(2.5) = -8.5 * cos((2π / 3)(2.5) - 2π / 3 * 1.3) - 35.5 H(2.5) = -8.5 * cos((5π / 3) - (2.6π / 3)) - 35.5 H(2.5) = -8.5 * cos(2.4π / 3) - 35.5Final Vertical Distance: We calculate the cosine value and the final height. H(2.5) = -8.5 * cos(0.8π) - 35.5 H(2.5) = -8.5 * (-0.5877852523) - 35.5  (using a calculator for cos(0.8π)) H(2.5) = 4.996679642 - 35.5 H(2.5) = -30.50332036 cm Rounded to two decimal places, the vertical distance is -30.50 cm.

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