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The amount of daylight a particular location on Earth receives on a given day of the...

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oyechalphut | Student, Grade 10 | Honors

Posted May 22, 2013 at 12:49 AM via web

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The amount of daylight a particular location on Earth receives on a given day of the year can be modelled by a sinusoidal function. The amount of daylight that Ancaster will experience in 2007 can be modelled by the function D(t) = 12.18 + 3.1 sin(0.017t – 1.376), where t is the number of days since the start of the year.

  1. On January 1, how many hours of daylight does Ancaster receive?
  2. What would the slope of this curve represent?
  3. The summer solstice is the day on which the maximum amount of daylight will occur. On what day of the year would this occur?
  4. Verify this fact using the derivative.
  5. What is the maximum amount of daylight Ancaster receives?
  6. What is the least amount of daylight Ancaster receives?

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted May 22, 2013 at 1:22 AM (Answer #1)

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Given the following function modelling the amount of sunlight on a particular day:

`D(t)=12.18+3.1sin(0.017t-1.376)`

The derivative is:

`D'(t)=.0527cos(0.017t-1.376)`

(1) `D(0)=12.18+3.1sin(0.017(0)-1.376)~~9.1386` so there will be approximately 9.1 hours of sunlight on Jan. 1

** The origin of the graph is taken to be Jan. 1 unless otherwise specified.**

(2) The slope of the curve at a point indicates the rate of change in the amount of sunlight on that day. A positive slope indicates days are getting longer, while a negative slope indicates days are getting shorter.

(3) The summer solstice is between  Jun. 20-22

(4) To find the maximum, set the derivative equal to zero:

`.0527cos(.017t-1.376)=0`

`==> cos(.017t-1.376)=0`

`==> .017t-1.376=cos^(-1)(0)=1.571`

`==>.017t=2.947`

`==>t~~173.341`

so the maximum daylight hours will occur on day 173 which will be June 21.

(5)(6) There are two ways to find the maximum and minimum. From trigonometry, we know that a sinusoid of the form `y=k+asin(b(x-h))` has a maximum at k+a, and a minimum at k-a. So the maximum will be 12.18+3.1=15.28 hours while the minimum will be 12.18-3.1=9.08 hours.

Using calculus, we find where the derivative equals zero. This occurs at `t=(1.376+(pi/2+kpi))/.017,k in ZZ` . There is a maximum at `t~~173` and the value of the function is 15.28. There is a minimum at `t~~358` and the value of the function there is 9.08

The graph:

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