Final answer:
The correct equation for the height of the initial point after t seconds is h = 8 cos (40πt) + 7, considering the paddle wheel's radius, the rate of revolution, and the maximum depth underwater.
Step-by-step explanation:
The paddle wheel of a boat measures 16 feet in diameter and is revolving at a rate of 20 revolutions per minute. The maximum depth of the paddle wheel under water is 1 foot. In order to determine the equation for the height of the initial point after t seconds, we'll have to consider the motion of the paddle wheel which can be modeled using sinusoidal functions, specifically the cosine function for a vertical circular motion.
Given:
- The diameter of the paddle wheel is 16 feet, so the radius (r) is half of that, i.e., 8 feet.
- The wheel makes 20 revolutions per minute, hence it completes one revolution in 3 seconds (t).
- The maximum depth under water is 1 foot which means the wheel dips 8 - 1 = 7 feet below the water level at the lowest point.
The equation for the height h of the point after t seconds in terms of cosine is:
h = 8 cos (40πt) + 7
where 40π is the angular velocity in radians per second (20 revolutions x 2π radians/revolution x 1 minute/60 seconds).