Could someone explain how to solve this math problem?
I've graphed it, but I'm not sure how to solve the rest of the questions.
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
b) Over the course of a single year, about how much does carbon dioxide concentration vary?
c) How much time is required for one complete cycle of carbon dioxide levels?
d) When during the year do the maximum and minimum occur?
NOTE - the formula should be:
H(t) = 0.013t^2 + 0.81t + 316 + 3.5 sin 2πt
since you have t^2 involved, the amount of variation depends on the year, but will always be roughly 2*3.5 = 7
since you have sin(2pi t) the period is 2pi/2pi = 1 year
max in April, min in Sep.
Don't know what graphing tool you use, but if you go to
http://rechneronline.de/function-graphs/
and enter
0.013x^2 + 0.81x + 316 + 3.5 *sin(2*pi*x)
for your function
with 0<=x<=5 and 300<=y<=350, the graph makes this quite clear.
how did you determine the maximum and the minimum?
To solve the math problem, you need to follow the steps below:
1. Graphing the function:
To graph the function H(t) = 0.013t^2 + 0.81t + 316 + 3.5sin(2πt) for the period 1960-1962, you will need a graphing tool or software. You can use online graphing calculators or graphing software like Microsoft Excel or Desmos to plot the graph. Here's how to do it:
- Set the x-axis to represent the time range from 0 to 3 since the period is 1960-1962.
- Set the y-axis to represent the CO2 concentration range from 300 to 340 since it is given in the question.
- Substitute the values of t into the equation and calculate H(t) for each value of t.
- Plot the points (t, H(t)) on the graph using the calculated values.
- Connect the points to form a smooth curve.
2. Variation of CO2 concentration over a single year:
To determine the amount by which the carbon dioxide concentration varies over a single year, you need to find the maximum and minimum values of H(t) within one year. Since the function is periodic, one year corresponds to one complete cycle. Here's what you need to do:
- Look for the maximum and minimum points of the graph within one year.
- Subtract the minimum value of H(t) from the maximum value of H(t) to find the range or variation of the carbon dioxide concentration over a single year.
3. Time required for one complete cycle:
To find the time required for one complete cycle of carbon dioxide levels, you need to identify the period of the sinusoidal term in the function H(t). Here's how you do it:
- The sinusoidal term can be written as 3.5sin(2πt).
- The coefficient of t in the sine function, 2π, determines the period.
- The period of a sine function is given by the formula T = 2π/|b|, where b is the coefficient of t.
- Calculate the period using the formula to find the time required for one complete cycle of carbon dioxide levels.
4. Maximum and minimum occurrence during the year:
To find when the maximum and minimum points occur during the year, you need to determine the values of t that correspond to these points. Here's what you need to do:
- Locate the maximum and minimum points on the graph.
- Check the corresponding values of t for these points.
- These values of t represent the times during the year when the maximum and minimum carbon dioxide concentrations occur.
By following these steps, you should be able to solve each part of the math problem and provide the appropriate explanations.