The concentration in parts per million of carbon dioxide a g

The concentration (in parts per million) of carbon dioxide (a green house gas) has been measured in Mauna Loa, Hawaii since 1959. The concentrations are known to have increased quadratically. The following table lists readings for three years. (Data from: National Oceanic and Atmospheric Administration.) Find a quadratic function f(x) = ax^2 + bx + c that gives the concentration in year x. Let x = 0 correspond to the year 1959, so the table represents the data points (1, 317), (30, 353), and (53, 394) Use this model to estimate the carbon dioxide concentration in the year 2019.

Solution

(a) Let us determine the coefficients a, b, and c in the quadratic function f(x).

Putting the values of x and f(x) from the given table we have

a(1)² + b(1) + c = 317, or, a + b + c = 317

  a(30)² + b(30) + c = 353, or, 900a + 30b + c = 353, and

  a(53)² + b(53) + c = 317, or, 2809a + 53b + c = 394

Subtracting first elation from the next two relations we get,

899a + 29b = 36 and

2808a + 52b = 77

Solving above, we get a = 0.010408 b = 0.918723

Substituting these values in first equation we get c = 316.070868

Thus, the quadratic equation is: f (x) = (0.010408) x² + (0.918723) x + 316.070868

(b) Using above relation to estimate the carbon dioxide concentration in the year 2019 is

f (2019 - 1959) = f (60) = (0.010408) 60² + (0.918723) 60 + 316.070868

= 400.4199

= 400.42