BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 1.3, Problem 25E
To determine

To find: The function that models the number of hours of daylight in terms of time at New Orleans and tests its accuracy.

Expert Solution

Answer to Problem 25E

The function that models the number of hours of daylight in terms of time at New Orleans is L(t)=12+2sin(2π365(t80)).

The accuracy of the model value from the actual value is approximately less than 1%.

Explanation of Solution

Given:

The latitude of New Orleans is 30°N and on March 31 the sunrise time was 5:51 am and sunset time was 6:18 pm.

Calculation:

From the give figure, the daylight lasts average 14 hours in the month of June and average 12 hours in the month of September at 30°N that is in New Orleans.

So that the amplitude of the curve is,

a=1412=2

Thus, the amplitude of the curve is a=2 days.

Since there are 365 days in a year, the horizontal stretching factor is 2π365.

Note that March 31st is the 90th day of the year and on that day the daylight was for 12hours and 27 minutes, that is 12.45 hours.

That is actual value of the daylight is 12.45 hours.

The number of the daylight in New Orleans on the t th day of the year is L(t)=12+2sin(2π365(t80)).

Substitute t=90 in the formula for L(t)

L(t)=12+2sin(2π365(9080))=12+2sin(2π365(10))=12+2sin(0.172)=12+0.34=12.34

Thus, the calculated value of the daylight is 12.34 hours.

From the above calculations, it is clear that the calculated value is different from the actual value.

The accuracy of the calculated value from the actual value is,

Actual value Model valueActual value=12.4512.3412.45=0.1112.450.009<1%

Therefore, the accuracy of the model value from the actual value is approximately less than 1%.

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