Trigonometry question

Question

As a dolphin swims along the side of a curise ship, it jumps to a height of four meters about the water surface and dives to a depth of four meters below the surface. It does this in a regular motion. A passenger notices that the dolphin completes each cycle in exactly 8 seconds (starting at the surface of the water).

(a) Describe the displacement of the dolphin relative to the water surface using a sinusoid curve and sketch the graph of the dolphin’s displacement relative to the water surface. You can use desmos to sketch the graph. (b) After 5 seconds, what is the dolphin’s position relative to the surface of the water?

(c) When is the dolhpin the furthest away from the water surface and when is it the deepest? (Hint: Remember, this occurs are multiple times.)

(d) What are some drawbacks of this model?

Answer 1

period = 8

amplitude = 4

midline = 0

no phase shift

A= amplitude=4

period = 2pi/B

B = 2pi/8 = pi/4

D=midline = 0

C= phase shift = 0, if the dolphin starts at water surface level heading up

h(t) = Asin(B+C)+D

= 4sin[(pi/4)t]

in 8 seconds the dolphin's position is h(8) = 4sin(8pi/4) = 4sin2pi = 4(0)= 0 = surface level

in 5 seconds the position is h(5) = 4sin(5pi/4) = 4(-sqr2/2) = about -4(.707) = -2.828 feet below the water surface

It's furthest from the water surface when it's 4 feet above or 4 feet below

when h(t)=4 or -4 = 4sin[(pi/4)t]

sin^-1[(pi/4)t] = +/- 1

2, 10, 18, 26, ... seconds for maximum distance, 4 feet above the water surface

it's deepest, 4 feet below, at 6, 14, 22, 30, ... seconds

it's at the water surface at 0, 4, 8, 12, 16, 20, 24, ... seconds

the model assumes the dolphin never slows down or speeds up, regardless of water currents or any obstacles or how tired it gets