# 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...

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.

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

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### 1 Answer

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: