Help Derivatives - maximizing sunlight through a window

[livestrong136]

Help Derivatives ASAP -- maximizing sunlight through a window

3. The amount of daylight a particular location on Earth receives on a given day of the year can be modeled by a sinusoidal function. The amount of daylight that Windsor, Ontario will experience in 2007 can be modeled 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.


c. The summer solstice is the day on which the maximum amount of daylight will occur. On what day of the year would this occur?

It happens when sin(0.017t – 1.376) = 1 since other constants won't change with change in t, only this sin function will change with change in t and the maximum value t hat a sine function can get is at pi/2 ie sin pi/2 = 1
i.e 0.017t-1.376 = pi/2

So how should I calculate the t from this equation

d. Verify this fact using the derivative.
Verify it by taking derivative,
d[D(t)]/dt = 3.1*0.01(cos(0.01t-1.376) = 0
Which implies that cos(0.01t-1.376) = 0
ie 0.017t-1.376 = pi/2 ..Same condition as we got in part c

My reasoning that d'(t) = 0 is correct, but rest looks wrong.

e. What is the maximum amount of daylight Windsor receives?
Maximum amount of daylight can be found out by putting t obtained from parts (c) or (d) in the equation. How would we do that.

 

[tt2348]

how do you mean it looks wrong? and you said yourself sin is max at 0.017t-1.376 = pi/2, where sin is 1. so for an equation that looks like A+Bsin(Ct+F), plugging in Ct+F=pi/2 =>
D(tmax)=A+B*sin(pi/2)
solving for t won't give you an integer number of days ( but the question asks what day.. so round down)

 

[SammyS]

3. The amount of daylight a particular location on Earth receives on a given day of the year can be modeled by a sinusoidal function. The amount of daylight that Windsor, Ontario will experience in 2007 can be modeled 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.

c. The summer solstice is the day on which the maximum amount of daylight will occur. On what day of the year would this occur?

It happens when sin(0.017t – 1.376) = 1 since other constants won't change with change in t, only this sin function will change with change in t and the maximum value t hat a sine function can get is at pi/2 ie sin pi/2 = 1
i.e 0.017t-1.376 = pi/2

So how should I calculate the t from this equation

...

Use algebra to solve for t.

The approximate answer is 173.341.

I suppose t is an integer, so round that to the nearest integer.

 

[livestrong136]

thnx a lot SammyS, can someone help me with d and e.

 

[Infinitum]

For (d) What you've done seems correct. The same condition verifies the fact.

For (e) Use the 't' you get from (c) in the D(t) equation to get the maximum daylight.