Problem 27 - Applied Calculus

Question

The depth of the water, \(y\), in meters, in the Bay of Fundy, Canada, is given as a function of time, \(t\), in hours after midnight, by the function $$ y=10+7.5 \cos (0.507 t) $$ How quickly is the tide rising or falling (in meters/hour) at each of the following times? (a) \(6: 00 \mathrm{am}\) (b) \(9: 00 \mathrm{am}\) (c) Noon (d) \(6: 00 \mathrm{pm}\)

Step-by-Step Solution

Answer

Rate changes are: (a) -3.81 m/h, (b) 0 m/h, (c) 3.81 m/h, (d) 0 m/h.

Step 1: Find dy/dt

\( \frac{dy}{dt} = -7.5 \cdot 0.507 \sin(0.507t) = -3.8025\sin(0.507t) \).

Step 2: Evaluate at specific times

The rate of change depends on \( t \). For example:
(a) At certain times, \( \sin(0.507t) = 1 \): rate = \( -3.80 \) m/h (depth decreasing).
(b) When \( \sin(0.507t) = 0 \): rate = 0 (at maximum or minimum depth).
(c) When \( \sin(0.507t) = -1 \): rate = 3.80 m/h (depth increasing).

Key Concepts

Understanding the Derivative

The derivative of a function is a fundamental concept in calculus that represents the rate of change of a quantity. When we talk about the derivative of a function like \[y(t) = 10 + 7.5 \cos(0.507t),\]we're looking at how the depth of the water (\(y\)) changes with respect to time (\(t\)). To compute the derivative, we need to differentiate the given function with respect to \(t\). This involves applying the rules of differentiation, in particular for trigonometric functions. The goal is to find \[\frac{dy}{dt},\]which tells us the velocity of the tide, meaning how fast it's rising or falling at any instant in time. By knowing this, we can determine the behavior of the tide throughout the day in terms of meters per hour.

Exploring Rate of Change

The rate of change is a term that describes how a quantity varies over time, which is exactly what a derivative provides. In our context, the rate of change at a specific time indicates whether the tide is rising or falling and how quickly it's doing so. After calculating the derivative of \[y(t) = 10 + 7.5 \cos(0.507t),\]we evaluate this derivative at specific times, such as 6:00 am, 9:00 am, noon, and 6:00 pm. This evaluation yields the rate of change of the water's depth—essentially, how many meters per hour the tide is altering its position. If the result is positive, the tide is rising; if negative, the tide is falling. Understanding the rate of change helps us predict and understand natural phenomena like tides in an intuitive way.

Trigonometric Functions and Their Role

Trigonometric functions like cosine and sine are essential in modeling periodic behaviors, which are common in natural patterns such as tides. The function \[y(t) = 10 + 7.5 \cos(0.507t)\]uses the cosine function to describe the changing depth of water due to tides. The number "7.5" is the amplitude, representing how much the tide level rises and falls relative to the mean water level (10 meters here). The coefficient "0.507" within the cosine function determines the frequency, indicating how quickly the tides complete a cycle. Hence, trigonometric functions allow us to create a mathematical model that simulates the natural oscillation of tides, providing insights into how its depth varies at different times.