   Chapter 8.4, Problem 72E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Tides Throughout the day, the depth of water D (in meters) at the end of a dock varies with the tides. The depth for one particular day can be modeled by D = 3.5 + 1.5   cos π t 6 ,   0 ≤ t ≤ 24 where t is the time (in hours), with t = 0 corresponding to midnight.(a) Determine dD/dt.(b) Evaluate dD/dt for t = 4 and t = 20, and interpret the meaning of these values in the context of the problem.(c) Find the time(s) when the water depth is the greatest and the time(s) when the water depth is the least.(d) What is the greatest depth? What is the least depth?(e) When is the rate of change of the water depth the greatest? the least?

(a)

To determine

To calculate: The derivative of the depth of water for any particular day that is dDdt, when the depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24.

Explanation

Given Information:

The depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24.

Formula used:

Cosine derivative Rule:

ddx[cosu]=sinududx

General power derivative rule:

ddxxn=nxn1

Calculation:

Consider the provided trigonometric function:

D=3.5+1.5cosπt6

Differentiate the above function with respect to t using Cosine and Power derivative rules.

dDdt=ddx[3.5+1.5cosπt6]=ddx[3

(b)

To determine

To calculate: The values of dDdt at t=4 and t=20, when the depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24. And also interpret meaning of those values.

(c)

To determine

To calculate: The times when the depth of water is the greatest and the depth of water is the least, when the depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24.

(d)

To determine

To calculate: The greatest depth of water and the least depth of water, when the depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24.

(e)

To determine

To calculate: The times, at which the greatest rate of change of the depth of water will occur and the times, at which the least rate of change of the depth of water will occur, when the depth of water for any particular day is modeled as D=3.5+1.5cosπt6 for 0t24.

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