4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Equations and inequalities describe the relationship between two mathematical expressions.

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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

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Question

Chapter 5, Problem 10PS

To determine

Use a graphing utility to graph both models in the same viewing window. Use a viewing window of $0≤t≤365$

Expert Solution

Answer to Problem 10PS

Explanation of Solution

Given information: The number of hours of daylight that occur at any location on Earth depends on the time of year and the latitude of the location. The equations below model the numbers of hours of daylight in Seward, Alaska ( $60°$ latitude), and New Orleans, Louisiana ( $30°$ latitude).

$D=12.2−6.4cos[π(t+0.2)182.6]$ Seward

$D=12.2−1.9cos[π(t+0.2)182.6]$ New Orleans

In these models, $D$ represents the number of hours of daylight and t represents the day, with $t=0$ corresponding to January $1$ .

Calculation:

Let us consider the numbers of hours of daylight in Seward, Alaska ( $60°$ latitude), and New Orleans, Louisiana ( $30°$ latitude) plotted against in a graph below,

EBK PRECALCULUS W/LIMITS, Chapter 5, Problem 10PS

Hence, the graph is shown above.

To determine

Find the days of the year on which both cities receive the same amount of daylight.

Expert Solution

Answer to Problem 10PS

April $1st$ and October $2nd$ of a leap year.

Explanation of Solution

Given information:

The number of hours of daylight that occur at any location on Earth depends on the time of year and the latitude of the location. The equations below model the numbers of hours of daylight in Seward, Alaska ( $60°$ latitude), and New Orleans, Louisiana ( $30°$ latitude).

$D=12.2−6.4cos[π(t+0.2)182.6]$ Seward

$D=12.2−1.9cos[π(t+0.2)182.6]$ New Orleans

In these models, $D$ represents the number of hours of daylight and t represents the day, with $t=0$ corresponding to January $1$ .

Calculation:

The two regions receive same amount of daylight on only two days of the year as can be seen from the above plot.

The first day on which the daylight received is same is the $91st$ day of the year that is, on the April $1st$ of a leap year, both the regions receive same daylight.

The second day on which the daylight received is same is the $74th$ day of the year that is, on the October $2nd$ of a leap year, both the regions receive same daylight.

Hence, on April $1st$ and October $2nd$ of a leap year, both the regions receive same daylight.

To determine

Which city has the greater variation in the number of hours of daylight? Which constant in each model would you use to determine the difference between the greatest and least numbers of hours of daylight?

Expert Solution

Answer to Problem 10PS

Seward.

$3.8$

$12.8$

Explanation of Solution

Given information

$D=12.2−6.4cos[π(t+0.2)182.6]$ Seward

$D=12.2−1.9cos[π(t+0.2)182.6]$ New Orleans

In these models, $D$ represents the number of hours of daylight and t represents the day, with $t=0$ corresponding to January $1$ .

Calculation:

Seward experiences a greater variation in daylight hours tha New orleans .

The numbers of hours of daylight in Seward, Alaska is expressed as,

$D=12.2−6.4cos[π(t+0.2)182.6]$

The numbers of hours of daylight in New Orleans, Louisiana is expressed as,

$D=12.2−1.9cos[π(t+0.2)182.6]$

Since the varying part in the above expressions is trigonometric part, so,the contants that should be used to calculate the difference between the greatest and the least number of daylight hours is the coefficient of the trigonometric functions.

Since the value of a cosine function ranges from $−1$ to $1$ , so the difference between the greatest and the least number of daylight in Seward are given as below,

$(12.2+6.4)−(12.2−6.4)=12.8$

But the difference between the greatest and least number of daylight in New Orleans, are as given below

$(12.2+1.9)−(12.2−1.9)=3.8$

Hence, Seward has greater variation and the greatest difference is $12.8$ while the least difference is $3.8$ .

To determine

Determine the period of each model.

Expert Solution

Answer to Problem 10PS

$P1=365.2$

$P2=365.2$

Explanation of Solution

Given information

$D=12.2−6.4cos[π(t+0.2)182.6]$ Seward

$D=12.2−1.9cos[π(t+0.2)182.6]$ New Orleans

In these models, $D$ represents the number of hours of daylight and t represents the day, with $t=0$ corresponding to January $1$ .

Calculation:

The period $P1$ for the expression the numbers of hours of daylight in Seward can be calculated by determining the reciprocal of the coefficient of $t$ in the argument of the given cosine function of the expression and multiplying it by $2π$ as follows,

$P1=1π182.6×2π=2π×182.6π=365.2$

The period $P2$ for the expression the numbers of hours of daylight in New Orleans can be calculated by determining the reciprocal of the coefficient of $t$ in the argument of the given cosine function of the expression and multiplying it by $2π$ as follows,

$P2=1π182.6×2π=2π×182.6π=365.2$

Hence $P1=365.2$ , $P2=365.2$

Chapter 5, Problem 9PS

Chapter 5, Problem 11PS

Chapter 5 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E

Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - Prob. 56E Ch. 5.1 - Prob. 57E Ch. 5.1 - Prob. 58E Ch. 5.1 - Prob. 59E Ch. 5.1 - Prob. 60E Ch. 5.1 - 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Use a graphing utility to graph both models in the same viewing window. Use a viewing window of 0 ≤ t ≤ 365

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Use a graphing utility to graph both models in the same viewing window. Use a viewing window of 0≤t≤365<m:math><m:mrow><m:mn>0</m:mn><m:mo>≤</m:mo><m:mtext>t</m:mtext><m:mo>≤</m:mo><m:mtext>365</m:mtext></m:mrow></m:math>

Answer to Problem 10PS

Explanation of Solution

Find the days of the year on which both cities receive the same amount of daylight.

Answer to Problem 10PS

Explanation of Solution

Which city has the greater variation in the number of hours of daylight? Which constant in each model would you use to determine the difference between the greatest and least numbers of hours of daylight?

Answer to Problem 10PS

Explanation of Solution

Determine the period of each model.

Answer to Problem 10PS

Explanation of Solution

Chapter 5 Solutions

Use a graphing utility to graph both models in the same viewing window. Use a viewing window of $0≤t≤365$