Some of the highest tides in the world occur in the Bay of...

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at $ 6:45 AM. $ This helps explain the following model for the water depth $ D $ (in meters) as a function of the time $ t $ (in hours after midnight) on that day: $ D(t) = 7 + 5 \cos [0.503(t - 6.75)] $ How fast was the tide rising (or falling) at the following times? (a) 3:00 $ AM $ (b) 6:00 $ AM $ (c) 9:00 $ AM $ (d) Noon

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at $ 6:45 AM. $ This helps explain the following model for the water depth $ D $ (in meters) as a function of the time $ t $ (in hours after midnight) on that day:

$ D(t) = 7 + 5 \cos [0.503(t - 6.75)] $
How fast was the tide rising (or falling) at the following times?

(a) 3:00 $ AM $
(b) 6:00 $ AM $
(c) 9:00 $ AM $
(d) Noon

Key Concepts

Sinusoidal Models in Tidal Analysis

Sinusoidal functions, such as sine and cosine, are widely used to model periodic phenomena like tides. They capture essential features including amplitude, which represents the maximum deviation from the mean level, the period which indicates how long one complete cycle takes, and phase shifts that align the model with specific time events. These models reflect natural oscillations such as the rise and fall of sea levels.

Application of the Chain Rule in Differentiation

When differentiating composite functions, such as a cosine function with a linear transformation in its argument, the chain rule is essential. It allows for the calculation of the derivative by multiplying the derivative of the outer function (cosine, in this case) by the derivative of the inner function (the linear transformation of time). This results in an expression for the rate of change of the tide's water depth with respect to time.

Physical Interpretation of Derivatives

In the context of tidal motion, the derivative of the water depth function represents the instantaneous rate at which the tide is rising or falling. A positive derivative indicates an increasing water level (rising tide), while a negative derivative indicates a decreasing water level (falling tide). This interpretation is key to understanding and predicting the behavior of tides at specific times.

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