Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
1st Edition
ISBN: 9780078682278
Author: McGraw-Hill, Berchie Holliday
Publisher: Glencoe/McGraw-Hill
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Chapter 6.6, Problem 16E

a.

To determine

Find the amount of daylight hours for the middle of each month.

a.

Expert Solution
Check Mark

Answer to Problem 16E

  MonthSunriseA.M.SunsetP.M.Day  lighthoursJanuary7:194:479:28february6:565:2410:28March6:165:5711:41April5:256:2913:04May4:447:0114:17June4:247:2615:02July4:337:2814:55August5:017:0114:00September5:316:1412:43October6:015:2411:23November6:364:4310:07December7:084:289:20

Explanation of Solution

Given information:

The table contains sunrise and sunset time; 1 represent the middle of January 2 represent the middle of February and so on.

  MonthSunriseA.M.SunsetP.M.January7:194:47february6:565:24March6:165:57April5:256:29May4:447:01June4:247:26July4:337:28August5:017:01September5:316:14October6:015:24November6:364:43December7:084:28

Calculation:

The table shows the amount of daylight hours for the middle of each month.

  MonthSunriseA.M.SunsetP.M.Day  lighthoursJanuary7:194:479:28february6:565:2410:28March6:165:5711:41April5:256:2913:04May4:447:0114:17June4:247:2615:02July4:337:2814:55August5:017:0114:00September5:316:1412:43October6:015:2411:23November6:364:4310:07December7:084:289:20

b.

To determine

Find the amplitude of sinusoidal function that model the daylight hours.

b.

Expert Solution
Check Mark

Answer to Problem 16E

  2:56 hours

Explanation of Solution

Given information:

The table contains sunrise and sunset time; 1 represent the middle of January 2 represent the middle of February and so on.

  MonthSunriseA.M.SunsetP.M.January7:194:47february6:565:24March6:165:57April5:256:29May4:447:01June4:247:26July4:337:28August5:017:01September5:316:14October6:015:24November6:364:43December7:084:28

Calculation:

Consider the table,

  MonthSunriseA.M.SunsetP.M.Day  lighthoursJanuary7:194:479:28february6:565:2410:28March6:165:5711:41April5:256:2913:04May4:447:0114:17June4:247:2615:02July4:337:2814:55August5:017:0114:00September5:316:1412:43October6:015:2411:23November6:364:4310:07December7:084:289:20

Amplitude is the half of difference between highest and lowest temperature,

  A=15:02−9:202A=5:422A=2:56 hours

Hence amplitude is 2:56 hours .

c.

To determine

Find the vertical shift of sinusoidal function that model the day light hours.

c.

Expert Solution
Check Mark

Answer to Problem 16E

  12:11 hours

Explanation of Solution

Given information:

The table contains sunrise and sunset time; 1 represent the middle of January 2 represent the middle of February and so on.

  MonthSunriseA.M.SunsetP.M.January7:194:47february6:565:24March6:165:57April5:256:29May4:447:01June4:247:26July4:337:28August5:017:01September5:316:14October6:015:24November6:364:43December7:084:28

Calculation:

The vertical shift is average of highest and lowest temperature,

  h=15:02+9:202h=24:222h=12:11 hours

Hence, the vertical shift of sinusoidal function that model the day light hours is 12:11 hours .

d.

To determine

Find the period of sinusoidal function that model the day light hours.

d.

Expert Solution
Check Mark

Answer to Problem 16E

  12

Explanation of Solution

Given information:

The table contains sunrise and sunset time; 1 represent the middle of January 2 represent the middle of February and so on.

  MonthSunriseA.M.SunsetP.M.January7:194:47february6:565:24March6:165:57April5:256:29May4:447:01June4:247:26July4:337:28August5:017:01September5:316:14October6:015:24November6:364:43December7:084:28

Calculation:

The period is 12 now the value of k will be,

  p=2πk12=2πkk=π6

Hence, period is 12 .

e.

To determine

Write the sinusoidal function that model the day light hours.

e.

Expert Solution
Check Mark

Answer to Problem 16E

  y=2.93sin(π6t−1.7260)+12.18

Explanation of Solution

Given information:

The table contains sunrise and sunset time; 1 represent the middle of January 2 represent the middle of February and so on.

  MonthSunriseA.M.SunsetP.M.January7:194:47february6:565:24March6:165:57April5:256:29May4:447:01June4:247:26July4:337:28August5:017:01September5:316:14October6:015:24November6:364:43December7:084:28

Calculation:

The standard form of sinusoidal function is,

  y=Asin(kt+c)+h

Substitute the given values,

  y=2.93sin(π6t+c)+12.18

  put  t=1  and  y=9.46  to  find  c,9.46=2.93sin(π6(1)+c)+12.189.46−12.182.93=sin(π6+c)arc sin(−0.928)=0.5236+cc=−0.189−0.5236c=−1.71260now.y=2.93sin(π6t−1.7260)+12.18

Hence, y=2.93sin(π6t−1.7260)+12.18

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Chapter 6.6, Problem 15E
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Chapter 6.6, Problem 17E

Chapter 6 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

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