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The number of hours of daylight in Vancouver can be modelled by a
sinusoidal function of time, in days. The longest day of the year is
June 21, with 15.7 h of daylight. The shortest day of the year is
December 21, with 8.3 h of daylight.
a) Find an equation for the number of hours of daylight on the
nth day of the year.
b) Use your equation to predict the number of hours of daylight in
Vancouver on January 30th

Sagot :

Answer:

a) Finding the Equation

1. Determine the midline: The average of the maximum and minimum daylight hours gives the midline of the sinusoidal function. (15.7 hours + 8.3 hours) / 2 = 12 hours

2. Determine the amplitude: The amplitude is the distance from the midline to the maximum (or minimum) value. 15.7 hours - 12 hours = 3.7 hours

3. Determine the period: The period is the length of one complete cycle, which is one year or 365 days.

4. Determine the phase shift: We’ll use June 21st (the longest day) as our starting point. Since the sine function starts at its midline and increases, we need to shift the graph to the right. June 21st is approximately the 172nd day of the year.

5. Construct the equation: We’ll use the following general form for a sinusoidal function:

y = a * sin(b(x - c)) + d

Where:

‘y’ is the number of hours of daylight

‘x’ is the day of the year

‘a’ is the amplitude (3.7 hours)

‘b’ is the frequency (2π/period = 2π/365)

‘c’ is the phase shift (172 days)

‘d’ is the midline (12 hours)

Therefore, the equation is:

y = 3.7 * sin((2π/365)(x - 172)) + 12

b) Predicting Daylight Hours on January 30th

Determine the day number: January 30th is the 30th day of the year.

Substitute into the equation:

y = 3.7 * sin((2π/365)(30 - 172)) + 12

y ≈ 8.5 hours

Explanation:

Therefore, we can predict that there will be approximately 8.5 hours of daylight in Vancouver on January 30th.

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