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The paddle wheel of a boat has a diameter of 16 feet and is rotating at a speed of 20 revolutions per minute. The maximum depth of the paddle wheel beneath the water surface is 1 foot. Assuming a point is positioned at the lowest point of the wheel at t=0, formulate an equation expressing the height of this initial point as a function of time (t) in seconds.

User Aramil Rey
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Final answer:

The height of the initial point on the paddle wheel can be modeled using a sine function. The formula for the height is h = r - R * sin(theta), where r is the maximum depth of the paddle wheel, R is the radius of the wheel, and theta is the angle of rotation.

Step-by-step explanation:

The height of the initial point on the paddle wheel can be modeled using a sine function since the wheel is rotating in a circular motion. The formula for the height of a point on a rotating wheel is h = r - R * sin(theta), where h is the height, r is the maximum depth of the paddle wheel (1 foot), R is the radius of the wheel (diameter/2), and theta is the angle of rotation in radians.

First, we need to convert the diameter of the wheel to the radius by dividing it by 2: R = 16 feet / 2 = 8 feet.

The time period for one revolution can be determined by taking the reciprocal of the rotational speed: T = 1 / (20 revolutions/minute) = 1 / (20/60) = 3 seconds/revolution. Since the height of the point on the wheel is a function of time, we can express theta in terms of time: theta = (2 * pi * t) / T, where t is the time in seconds.

Substituting these values into the formula, we get: h = 1 - 8 * sin((2 * pi * t) / 3).

User Andrew Coats
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