Question

Antonio's toy boat is bobbing in the water next to a dock. Antonio starts his stopwatch, and measures the vertical distance from the dock to the height of the boat's mast, which varies in a periodic way that can be modeled approximately by a trigonometric function.
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.

What is the vertical distance 2.5 seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places.

Answer 1

Answer:

Formula of the function is found:

$H(t) = 8.5cos(\frac{2\pi}{3}(t-1.3))-35.5$

For t= 2.5s , vertical distance is:

H(2.5) = +42.38cm , H(2.5) = -42.38cm

Explanation:

General form of trigonometric function H(t) is given by:

$H(t)=Acos(B(t-C))+D$

Where

A = Amplitude

Period = 3s = 2π/B

B = 2π/3

C = Phase shift = 1.3s

D = Vertical Shift

Find A (Amplitude):

Amplitude = (highest value - lowest value)/2

Amplitude = (-27-(-44))/2

Amplitude = (-27+44)/2

Amplitude = 8.5

Find D (Vertical Shift)

Vertical shift can be found by finding midpoint

Midpoint = (highest value + lowest value)/2

Midpoint = (-27-44)/2

Midpoint = -35.5

Substitute the values of A,B,C,D in the general form of trigonometric function.

$H(t)=Acos(B(t-C)) +D$

$H(t) = 8.5cos(\frac{2\pi}{3}(t-1.3))-35.5$

which is the formula of the function

For t = 2.5s

$H(2.5)=8.5cos(\frac{2\pi}{3}(2.5-1.3))-35.5 \\H(2.5)= +42.38 {cm} ,H(2.5)=-42.38cm$