Pre-Calculus, 9th edition
35

 PAGE: 426 SET: Exercises PROBLEM: 35

(a)

The number of hours of sunlight on the summer solstice of was .

The number of hours of sunlight on the winter solstice was .

Find the sinusoidal function of the form .

Where is amplitude and period is and phaseshift is .

The amplitude of the function is .

.

The amplitude of the function is .

The verticalshift of the fucntion is .

The verticalshift of the function is .

The days repeats every days, hence the time period for one cycle is  .

Therefore, .

To find the horizontal shift , divide the time period days into four subintervals of length .

Hence the subintervals are .

The function of the sine wave is increasing on the interval  and decreasing on the interval .

Hence a local maximum occurs at days.

But the maximum value of summer solstice occurs at days.

Therefore, the horizontal shift is .

hence .

Substitute , , , in .

.

Therefore, the sinusoidal function is .

(b)

Find the number of hours of daylight on April , the st day of the year.

Substitute in .

The number of hours to predict the daylight on April are  .

(c)

Graph:

Graph the function .

(d)

Graph:

Graph the function .

Observe the graph :

The actual number of hours of daylight on April are  is same as the predicted amount.

(a) The sinusoidal function is .

(b) The number of hours to predict the daylight on April are  .

(c)

Graph of the function is

(d)

Observe the graph :

The actual number of hours of  daylight on April are  is same as the predicted amount.

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