Daylight Hours Homework: Solving for Portland, OR 2016

  • Thread starter Coltjb7
  • Start date
In summary: I find a specific date where it is increasing/decreasing the most. She said we could use our calculators to do it, but I cannot remember how to do this on a calculator. I have the TI-84 Plus.We are not allowed to use calculators or computers in this course. I have to solve it algebraically. Can you help me understand how to do that?In summary, the given equation, H(t)=3.3sin(0.017214t-1.369247)+12, estimates the number of daylight hours in Portland, OR for any day t during 2016. To find the date when the number of daylight hours is increasing or decreasing the
  • #1
Coltjb7
15
0

Homework Statement


My teacher created this following problem:
The equation H(t)=3.3sin(0.017214t-1.369247)+12 estimates the number of daylight hours in Portland, OR for any day, t, during 2016.
1. Give the date in 2016 when the number of daylight hours is increasing the fastest. How many additional minutes of daylight do we gain on that day?
2.Give the date in 2016 when the number of daylight hours is decreasing the fastest. How many minutes of daylight do we lose on that day?
3. How many days into the year is the longest day of 2016? What is the date?

Homework Equations


H(t)=3.3sin(0.017214t-1.369247)+ 12 0<t January 1=0; December 31=365; *2016 is
H'(t)=0.0568062cos(0.017214t-1.369247) a leap year.
H''(t)=-9.77861927E^-4sin(0.017214t-1.369247)

The Attempt at a Solution


I attempted to use the first derivative test but am not sure what I am doing. Do I need to solve for t? One kid at my table thinks they figured it out and got t=(about)74 so the answer t #1 would be around March 10th. I m totally lost. Please help
 
Physics news on Phys.org
  • #2
Coltjb7 said:

Homework Statement


My teacher created this following problem:
The equation H(t)=3.3sin(0.017214t-1.369247)+12 estimates the number of daylight hours in Portland, OR for any day, t, during 2016.
1. Give the date in 2016 when the number of daylight hours is increasing the fastest. How many additional minutes of daylight do we gain on that day?
2.Give the date in 2016 when the number of daylight hours is decreasing the fastest. How many minutes of daylight do we lose on that day?
3. How many days into the year is the longest day of 2016? What is the date?

Homework Equations


H(t)=3.3sin(0.017214t-1.369247)+ 12 0<t January 1=0; December 31=365; *2016 is
H'(t)=0.0568062cos(0.017214t-1.369247) a leap year.
H''(t)=-9.77861927E^-4sin(0.017214t-1.369247)

The Attempt at a Solution


I attempted to use the first derivative test but am not sure what I am doing. Do I need to solve for t? One kid at my table thinks they figured it out and got t=(about)74 so the answer t #1 would be around March 10th. I m totally lost. Please help
Hello Coltjb7. Welcome to PF !

I general, what does the first derivative test tell you?
 
  • #3
SammyS said:
Hello Coltjb7. Welcome to PF !

I general, what does the first derivative test tell you?
It would tell me on what intervals the equation is increasing/decreasing.
Honestly I am having trouble finding the Critical Values.

Thank you for the welcome
 
  • #4
Coltjb7 said:
It would tell me on what intervals the equation is increasing/decreasing.
Honestly I am having trouble finding the Critical Values.

Thank you for the welcome
Why don't you show us what you've tried to find the critical values?
 
  • #5
SteamKing said:
Why don't you show us what you've tried to find the critical values?
so what I have been trying to do is this:
H'(t)=0.0568062cos(0.017214t-1.369247)
Since anything multiplied by 0 is equal to zero I ignored the constant leaving me with this:
0=cos(0.017214t-1.369247).
I know that cos=0 at pi/2 and 3pi/2 but am having trouble figuring out how to work this out.
 
  • #6
Coltjb7 said:
so what I have been trying to do is this:
H'(t)=0.0568062cos(0.017214t-1.369247)
Since anything multiplied by 0 is equal to zero I ignored the constant leaving me with this:
0=cos(0.017214t-1.369247).
I know that cos=0 at pi/2 and 3pi/2 but am having trouble figuring out how to work this out.
Do you know about inverse trigonometric functions?
 
  • #7
Haha, yes I know about them but have not used them in a while. Would I do this:
0=cos(0.017214t-1.369247)
cos^-1(0)=0.017214t-1.369247
[(cos^-1(0))+1.369247]/0.017214=t
t=170.79
 
  • #8
So would my intervals go from (-infinity,170.79) and (170.79, infinity)?
 
  • #9
Coltjb7 said:
Haha, yes I know about them but have not used them in a while. Would I do this:
0=cos(0.017214t-1.369247)
cos^-1(0)=0.017214t-1.369247
[(cos^-1(0))+1.369247]/0.017214=t
t=170.79
That's one solution. What's the other one?
Coltjb7 said:
So would my intervals go from (-infinity,170.79) and (170.79, infinity)?
It's not clear what these intervals are for.
 
  • #10
SteamKing said:
That's one solution. What's the other one?

It's not clear what these intervals are for.
I honestly am not sure how to find the other solution. Mind helping me out? I would be using the intervals for my first derivative test. our teacher has us create a table like this:
Interval ( we gather this from the critical values)
Test Value (any number you want within each interval)
Sign of f'(x) (test the value you chose as a test value to see if it is >0 or <0)
Conclusion (where we say if it is increasing/decreasing)
 
  • #11
##H(t)\ ## gives the hours of daylight on day ##\ t\,.##

The derivative, ##\ H'(t)\ ## gives the rate at which the ##\ H(t)\ ## is changing on day ##\ t\,.##

If you solve for the derivative being zero, that gives the day on which the hours of daylight is minimum or maximum. That's what you're asked to find in part 3.

To find out when the hours of daylight is changing at a maximum or minimum rate, you need to look at the derivative of the function that gives the rate of change.

Added in Edit:
The following solution is for part 3,
3. How many days into the year is the longest day of 2016?

Coltjb7 said:
Haha, yes I know about them but have not used them in a while. Would I do this:
0=cos(0.017214t-1.369247)
cos^-1(0)=0.017214t-1.369247
[(cos^-1(0))+1.369247]/0.017214=t
t=170.79

That's some day in late June, close to the day with most sunlight hours.
 
Last edited:
  • #12
I understand I need to look at the derivative but how do I find a specific date where it is increasing/decreasing the most. She said we could use our calculators to do it, but I cannot remember how to do this on a calculator. I have the TI-84 Plus.
 
  • #13
Coltjb7 said:
I understand I need to look at the derivative but how do I find a specific date where it is increasing/decreasing the most. She said we could use our calculators to do it, but I cannot remember how to do this on a calculator. I have the TI-84 Plus.
Look at the derivative of WHICH function?
 
  • #14
Coltjb7 said:
I honestly am not sure how to find the other solution. Mind helping me out?

You already told me the solutions:

Coltjb7 said:
I know that cos=0 at pi/2 and 3pi/2 but am having trouble figuring out how to work this out.

You have apparently worked out the value of t using π/2 as the argument for the cosine where its value is zero. What do you get for t if the argument is 3π/2?
 
  • #15
SteamKing said:
You already told me the solutions:
You have apparently worked out the value of t using π/2 as the argument for the cosine where its value is zero. What do you get for t if the argument is 3π/2?
See that's what I am not understanding. I did nothing with pi/2. I ignored that information and solved it using cos^-1
 
  • #16
SammyS said:
Look at the derivative of WHICH function?
The original function. H(t)=3.3sin(0.017214t-1.369247)+12
 
  • #17
Coltjb7 said:
The original function. H(t)=3.3sin(0.017214t-1.369247)+12
No. Setting the derivative of that to zero would give the date of max or min hours of daylight.

You already have the function and the first and second derivatives!
Coltjb7 said:
...

H(t)=3.3sin(0.017214t-1.369247)+ 12 where 0<t January 1=0; December 31=365; *2016 is a leap year.
H'(t)=0.0568062cos(0.017214t-1.369247)
H''(t)=-9.77861927E^-4sin(0.017214t-1.369247)
## H'(t)\ ## is the rate of change in the hours of daylight.

You need to find the max and min for this rate of change, not for the number of hours (not until part 3 anyway).

rate of change: ##\ H'(t)\ ##

number of hours: ##\ H(t)\ ##
 
  • #18
Coltjb7 said:
See that's what I am not understanding. I did nothing with pi/2. I ignored that information and solved it using cos^-1
But what value did you do the cos-1 of?
 
  • #19
Coltjb7 said:
See that's what I am not understanding. I did nothing with pi/2. I ignored that information and solved it using cos^-1
So now i solved H'(t) for zero when t=pi/2 and 3pi/2. These are my new answers:
when t=pi/2 there is a critical value at 108.7306
When t=3pi/2 there is a critical value at 36.24.

So, is day number 108.7306 (do i round to day 109 or keep it at 108?) the day where the most daylight is being added? And, is day 36.2435 the day where the the daylight is decreasing the fastest?
 
  • #20
Coltjb7 said:
So now i solved H'(t) for zero when t=pi/2 and 3pi/2. These are my new answers:
when t=pi/2 there is a critical value at 108.7306
When t=3pi/2 there is a critical value at 36.24.

So, is day number 108.7306 (do i round to day 109 or keep it at 108?) the day where the most daylight is being added? And, is day 36.2435 the day where the the daylight is decreasing the fastest?
I think you have things askew here.

If H'(t) = cos(0.017214t-1.369247), then you want to find t such that it makes H'(t) = 0. Since H'(t) is a cosine function, cosine is zero when its argument is π/2 or 3π/2, which means that (0.017214t-1.369247) = π/2 or 3π/2. You have to solve for t to get those values where H'(t) = 0.
 
  • Like
Likes Coltjb7
  • #21
If ##\ H'(t)=0\,,\ ## then ##\ H(t)\ ## is at a max or min.

If ##\ H''(t)=0\,,\ ## then ##\ H'(t)\ ## is at a max or min.
 
  • #22
SteamKing said:
I think you have things askew here.

If H'(t) = cos(0.017214t-1.369247), then you want to find t such that it makes H'(t) = 0. Since H'(t) is a cosine function, cosine is zero when its argument is π/2 or 3π/2, which means that (0.017214t-1.369247) = π/2 or 3π/2. You have to solve for t to get those values where H'(t) = 0.
Ohhh. Okay. Thank you. I was going about it wrong and set t=pi/2 and 3pi/2 then plugged that value into H'(t). Now I realize that I made a stupid mistake. I worked out those values and came up with:

When pi/2=0.017214t-1.369247 Then following the same premise for 3pi/2
0.017214t=pi/2 +1.369247 t=353.2959
0.017214t=(pi+2.73849)/2
t=(pi+2.73849)/(2(0.017214))
t=170.7937

*I got the value 2.73849 by creating a common denominator of 2*

Now, are those my intervals for my first derivative test, or are they the values I need to answer questions 1&2?
 
  • #23
Coltjb7 said:
Now, are those my intervals for my first derivative test, or are they the values I need to answer questions 1&2?
Go back & review all of the posts.
 
  • #24
SammyS said:
Go back & review all of the posts.
I think I have got it. Thank you guys very much!
 

1. How do daylight hours affect the productivity and sleep patterns of individuals in Portland, OR in 2016?

Daylight hours have a significant impact on the circadian rhythm of individuals, which can affect their productivity and sleep patterns. With longer daylight hours in the summer months, people tend to have more energy and are able to accomplish more tasks. On the other hand, shorter daylight hours in the winter can lead to feelings of fatigue and decreased productivity.

2. How does the latitude and longitude of Portland, OR affect the amount of daylight hours in 2016?

Portland, OR is located at a latitude of 45.52 degrees North and a longitude of 122.68 degrees West. This location results in longer daylight hours during the summer months and shorter daylight hours during the winter months. The tilt of the Earth's axis and its rotation around the sun also play a role in the amount of daylight received in different latitudes.

3. What is the average number of daylight hours in Portland, OR during 2016?

In 2016, the average number of daylight hours in Portland, OR was around 15 hours in the summer months (June-August) and 9 hours in the winter months (December-February). The exact number of daylight hours can vary slightly from year to year due to factors such as leap years and weather patterns.

4. How does Daylight Saving Time affect the daylight hours in Portland, OR in 2016?

Daylight Saving Time (DST) is observed in Portland, OR, where the clocks are set one hour ahead in the spring and one hour back in the fall. This results in longer daylight hours in the evening during DST, as the sun sets later, but shorter daylight hours in the morning. However, DST does not significantly alter the total amount of daylight hours received in Portland, OR during 2016.

5. Can the amount of daylight hours in Portland, OR in 2016 affect the behavior of plants and animals?

Yes, the amount of daylight hours can have a significant impact on the behavior of plants and animals in Portland, OR during 2016. Longer daylight hours in the summer months can lead to increased photosynthesis and growth in plants, while shorter daylight hours in the winter can cause plants to go dormant. Animals may also change their behavior in response to the amount of daylight, such as migrating or hibernating.

Similar threads

Calculus Assignment Help: Temperature & Daylight Equations Explained
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Comparing Daylight Increase in Philadelphia Using a Model
    • Calculus and Beyond Homework Help
Replies
7
Views
6K
Number of hours of daylight - Periodic functions.
    • Precalculus Mathematics Homework Help
Replies
19
Views
6K
Applications of Derivatives. Needs Checking
    • Calculus and Beyond Homework Help
Replies
4
Views
4K
Periodic Functions Homework: Daylight Hours
    • Precalculus Mathematics Homework Help
Replies
3
Views
2K
MHB Solving 2 Watches with 12 Hour Cycle Problem
    • General Math
Replies
3
Views
12K
How Does Daylight Vary by Month Using a Sinusoidal Function?
    • Precalculus Mathematics Homework Help
Replies
2
Views
2K
Solve Logistic Equation: Find P(65) | Homework Due Early
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Calculating Daylight Hours Using Trigonometry
    • Introductory Physics Homework Help
Replies
7
Views
2K
Finding Amplitude-simple harmonic motion
    • Introductory Physics Homework Help
Replies
8
Views
26K

Hot Threads

Recent Insights

Back
Top