# solving for t?

• Nov 25th 2012, 02:46 PM
nyago
solving for t (daylight hours problem)?
i'm trying to solve the following problem for t: 720 = 11.895 + 2.545 sin [(2pi/366)(t-80)]. my formula is using the format y = d + a cos [b(t-c)]

I found a similar problem/solution online but don't quite understand how they solved the steps in bold... could someone explain how they got those answers in more detail?:

In Philadelphia the number of hours of daylight on day t (where t is the number of days after Jan. 1) is modeled by the function
L(t)= 12+2.83sin(2pi/365(t-365))

A) Which days of the year have about 10 hours of daylight?
B) Which days of the year have more than 10 hours of daylight?

• math - drwls, Saturday, August 13, 2011 at 10:13pm(A) Solve
10 = 12 + 2.83 sin[(2*pi/365*(t-365)]

-2 = 2.83 sin[(2*pi*/365)(t-365)]
-0.7067 = sin[(2*pi*/365)(t-365)]
sin[(2*pi*/365)(t-365)] = -0.7848
Solve for t.

[(2*pi*/365)(t-365)] = -0.90244
Use the first value

t-365 = -52
t = 313 days

Oct 1 is day 303. So Oct 11 is one answer. The other day will be 52 days after the winter solstice, or about Feb 11.
• Nov 25th 2012, 03:16 PM
jakncoke
Re: solving for t?
if $720 = 11.895 + 2.54 sin(\frac{2\pi}{366(t-80)})$ is your equation. Sin(x) reaches a max value of 1. So there is no way your equality could be right.
• Nov 25th 2012, 03:27 PM
Plato
Re: solving for t?
Quote:

Originally Posted by jakncoke
if $720 = 11.895 + 2.54 sin(\frac{2\pi}{366(t-80)})$ is your equation. Sin(x) reaches a max value of 1. So there is no way your equality could be right.

@jakncoke
Correct LaTeX code makes a more professional post.

$$720 = 11.895 + 2.54 \sin\left(\frac{2\pi}{366(t-80)}\right)$$ gives $720 = 11.895 + 2.54 \sin\left(\frac{2\pi}{366(t-80)}\right)$
Click on the “go advanced” tab. On the toolbar you will see $\boxed{\Sigma}$ clicking on that give the LaTeX wraps, . The code goes between them.
• Nov 25th 2012, 03:35 PM
nyago
Re: solving for t?
..........
• Nov 25th 2012, 04:07 PM
skeeter
Re: solving for t?
Quote:

that is not how i wrote my equation.... it's 720 = 11.895 + 2.545 sin [(2pi/366)(t-80)]. my equation is modelling daylight hours, I need to discuss solving the inequality y(t) ≥ 720 for t

this is my original formula:
y= 11.895 +2.545 sin [(2pi/366)(t-80.5)
one more time ... no way the right side of the equation is > the left side.

maybe you should post the original problem as it is presented to you ... you've more than likely made a mistake in your set-up.
• Nov 25th 2012, 04:14 PM
nyago
Re: solving for t?
This is the original problem:
Daylight is seasonal (there are more daylight hours in the summer and fewer in the winter).
According to the Unites States Naval Observatory, the duration of daylight in Los Altos Hills each day
of 2012 is given in the accompanying table (6-Daylight Los Altos Hills 2012.pdf), also available at
Astronomical Applications Department.
Your job is to analyze y, the minutes of daylight in Los Altos Hills, as a function of t, the number of
days since the beginning of 2012. In particular, you should
a) Produce an appropriately-labeled and appropriately-scaled graph of the daylight function
covering a time period of at least three calendar years including 2012;
b) Find a formula for the daylight function using constants determined from the data;
e) In the context of daylight duration, discuss solving the inequality y(t) ≥ 720 for t

This is the formula I got for problem b: y= 11.895 + 2.545 sin [(2pi/366)(t-80)] using the format y= d + a sin [b (t-c)]

a= amplitude
b = wavelength
d = centerline
c = phase shift
• Nov 25th 2012, 04:31 PM
skeeter
Re: solving for t?

y= 11.895 + 2.545 sin [(2pi/366)(t-80)]

what are the units for these two numbers?
• Nov 25th 2012, 04:38 PM
nyago
Re: solving for t?
Amplitude:
A= 1/2 (14.44-9.35)
= 2.545

Centerline:
D = 1/2 (14.44+9.35)
= 2.545
• Nov 25th 2012, 04:39 PM
topsquark
Re: solving for t?
Another thought. You derived an equation to fit the data? I couldn't find data for this. How did you get your equation?

-Dan
• Nov 25th 2012, 04:46 PM
nyago
Re: solving for t?
The data was from the US Naval Observatory... I chose the maximum/minimum amount of daylight from a one year set of data. Have you done these types of problems before??
• Nov 25th 2012, 04:48 PM
skeeter
Re: solving for t?

Fyi, I believe your units are in hours ... the 720 is in minutes.
• Nov 25th 2012, 04:52 PM
nyago
Re: solving for t?
the data set was in hours and minutes. that 's how i wrote the units for 11.895 + 2.545 ... and I thought the 720 was in hours

so, what should i do? i'm confused now
• Nov 25th 2012, 04:59 PM
nyago
Re: solving for t?
should i have rounded up my numbers? i didn't do that , otherwise the high and low points were off a bit when i graphed with an online grapher (desmos)
• Nov 25th 2012, 05:05 PM
skeeter
Re: solving for t?

Quote:

Your job is to analyze y, the minutes of daylight in Los Altos Hills, as a function of t, the number of
days since the beginning of 2012. In particular, you should
a) Produce an appropriately-labeled and appropriately-scaled graph of the daylight function
covering a time period of at least three calendar years including 2012;
b) Find a formula for the daylight function using constants determined from the data;
e) In the context of daylight duration, discuss solving the inequality y(t) ≥ 720 for t
your amplitude and vertical shift was in hours, so you need to modify your equation since y is in minutes ...

y= 60(11.895) + 60(2.545) sin [(2pi/366)(t-80)]
• Nov 25th 2012, 05:21 PM
nyago
Re: solving for t?
Oh, I see!!