   Chapter 8.4, Problem 71E ### 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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# Biology Plants do not grow at constant rates during a normal 24-hour period because their growth is affected by sunlight. Suppose that the growth of a certain plant species in a controlled environment is modeled by h =   0.2 t +   0.03  sin  2 π t where h is the height of the plant (in inches) and t is the time (in days), with t = 0 corresponding to midnight of day 1. During what time of day is the rate of growth of this plant(a) the greatest?(b) the least?

(a)

To determine

To calculate: The time of day at which rate of growth of plant is greatest, when the growth of a certain plant species in controlled environment is modeled as h=0.2t+0.03sin2πt.

Explanation

Given Information:

The growth of a certain plant species in controlled environment is modeled as h=0.2t+0.03sin2πt.

Formula used:

Sine derivative Rule:

ddx[sinu]=cosududx

Cosine derivative Rule:

ddx[cosu]=sinududx

General power derivative rule:

ddxxn=nxn1

Condition for critical points:

dydx=f(x)=0

Calculation:

Consider the provided function:

h=0.2t+0.03sin2πt

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

dhdt=ddt[0.2t+0.03sin2πt]=0.2ddt[t]+0.03ddt[sin2πt]=0.2+0.03cos2πtddt[2πt]=0.2+0.03cos2πt[2π]

Further solve as;

dhdt=0.2+0.03(2π)cos2πt

Again, differentiate the above function with respect to t using Cosine and Power derivative rules as;

d2hdt2=ddt[0.2+0.03(2π)cos2πt]=ddt[0.2]+0.03(2π)ddt[cos2πt]=0+0.03(2π)(sin2πt)ddt[2πt]=0.03(2π)(sin2πt)[2π]

Further solve as;

d2hdt2=0.03(2π)2(sin2πt)

For maximum rate of growth apply d2hdt2=0 as;

0

(b)

To determine

To calculate: The time of day at which rate of growth of plant is least, when the growth of a certain plant species in controlled environment is modeled as h=0.2t+0.03sin2πt.

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