Could someone explain how to find which months during the year the maximum/minimum points occur for this problem occurs? I think it's around May and Oct, but I don't get how to determine that on the graph
The National Oceanic and Atmospheric Administration (NOAA) has been measuring atmospheric
carbon dioxide concentations (in parts per million) at Mauna Loa, Hawaii since 1958. The data
closely follow the pattern H(t) = 0.013t2 + 0.81t + 316 + 3.5 sin 2πt, where t = 0 represents the
year 1960. (Complete dataset available at ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/c…
1. Explore the CO2 concentration model for the period 1960 – 1962.
a) Graph H for 0 ≤ t ≤ 3 and 300 ≤ H ≤ 340
d) When during the year do the maximum and minimum occur?
The formula should actually be: H(t) = 0.013t2 + 0.81t + 316 + 3.5 sin 2πt
H(t) = 0.013t^2 + 0.81t + 316 + 3.5 sin 2πt
Ah, the formula should have an exponent!
I hope you are taking Calculus
H ' (t) = .026t + .81 + 7πcos (2πt)
= 0 for a max or min
7πcos (2πt) + .026t + .81 = 0
Wow, that's a tough one to solve
Let's try Wolfram, one of the best pages for math.
http://www.wolframalpha.com/input/?i=solve+7πcos+%282πt%29+%2B+.026t++%2B+.81+%3D+0
using the first two positive values of
t = .2559 and t = .744
we get
H(.2559) = appr 319.7
H(.744) = appr 312.7
so I would say : .2559(12) = 3 or March of 1960 has a max of 319.7
.744(12) = 8.9 or then end of August of 1960 has a min of 312.7
repeat by taking the next 4 positive solutions from Wolfram's webpage
Without the help of pages such as the link above, solving equations like this are very onerous. Methods such as Newton's Method work, but require lots of tedious steps and lots of patience.
I want you to image doing this 40 years ago, when we had not calculators or the aid of webpages, doing this with only pencil and paper. We did it.
ah, thanks... i'm only taking "liberal arts math", so have only done a little bit of calculus.
how did you get this part?
H(.2559) = appr 319.7
H(.744) = appr 312.7
I substituted .2559 and .744 into the original equation.
Make sure the calculator is set to radians, not degrees.
To determine when the maximum and minimum occur for this problem, you can follow these steps:
1. Calculate the derivative of the function: To find the maximum and minimum points, we need to find where the derivative of the function is equal to zero. The derivative of the function H(t) can be calculated using calculus.
2. Set the derivative equal to zero: Set the derivative of H(t) equal to zero and solve for t. This will give you the values of t at which the maximum and minimum points occur.
3. Determine the corresponding months: Once you have the values of t, you can determine the corresponding months by adding them to the base year of 1960. Each unit of t represents one year, so adding the values of t to 1960 will give you the corresponding year. From there, you can determine the corresponding month.
Let's go through these steps for the given function H(t) = 0.013t^2 + 0.81t + 316 + 3.5sin(2πt):
1. Calculate the derivative:
The derivative of H(t) is given by H'(t) = 0.026t + 0.81 + 3.5(2π)cos(2πt).
2. Set the derivative equal to zero:
0.026t + 0.81 + 3.5(2π)cos(2πt) = 0.
Unfortunately, solving this equation algebraically may be quite challenging due to the presence of the trigonometric function. Therefore, we can use numerical methods like approximation or graphing tools to estimate the values of t where H'(t) is approximately equal to zero.
3. Determine the corresponding months:
Once you have estimated the values of t where H'(t) is approximately equal to zero, you can determine the corresponding months by adding these values to the base year of 1960. For example, if you find t = 2.5 and t = 5.2, you would add these values to 1960 to get the corresponding years. From there, you can determine the corresponding months based on the numerical value of t.
In order to provide a more accurate determination of the maximum and minimum points, it would be helpful to have access to the complete dataset mentioned in the problem statement. With the complete dataset, you could perform a more precise analysis of the maximum and minimum points over the specified time period.