The average monthly temperatures in New Orleans, Louisiana, are
given in the following tables:
MonthJFMAMJ°C 16.5 18.3 21.8 25.6 29.2 31.9 MonthJASOND°C 32.6 32.4 30.3 26.4 21.3 18.0
What characteristics of the function correspond to the constants a, b, c, and d?
Determine the temperature function T(t). [Hint: Consider January to be Month 0, February to be Month 1, etc.]
Number the months 1-12 and then plug the data into a graphing system to get a 3rd degree polynomial in the form ax^3+bx^2+cx+d=f(x)
To determine the temperature function T(t) and the corresponding constants a, b, c, and d, we need to analyze the given tables and understand the pattern.
Looking at the tables, we can see that the months are divided into two sets: JFMAMJ and JASOND. Let's consider the temperature function for each set of months separately.
For the JFMAMJ set of months:
The temperatures given in the first table correspond to the months January to June (Month 0 to Month 5). Let's consider the months in this order: 0, 1, 2, 3, 4, 5.
The temperature values in this set of months can be represented using a linear function of the form T(t) = at + b, where 't' represents the month.
The values for this function can be determined as follows:
T(0) = 16.5 (temperature for January)
T(1) = 18.3 (temperature for February)
T(2) = 21.8 (temperature for March)
T(3) = 25.6 (temperature for April)
T(4) = 29.2 (temperature for May)
T(5) = 31.9 (temperature for June)
By substituting these values into the linear function, we can solve for 'a' and 'b':
16.5 = a(0) + b, which simplifies to b = 16.5
18.3 = a(1) + b, which simplifies to a + 16.5 = 18.3, so a = 18.3 - 16.5 = 1.8
Therefore, for the JFMAMJ set of months, the temperature function T(t) can be expressed as:
T(t) = 1.8t + 16.5
For the JASOND set of months:
The temperatures given in the second table correspond to the months July to December (Month 6 to Month 11). Let's consider the months in this order: 6, 7, 8, 9, 10, 11.
Similar to the previous set, the temperature values in this set of months can also be represented using a linear function.
The values for this function can be determined as follows:
T(6) = 32.6 (temperature for July)
T(7) = 32.4 (temperature for August)
T(8) = 30.3 (temperature for September)
T(9) = 26.4 (temperature for October)
T(10) = 21.3 (temperature for November)
T(11) = 18.0 (temperature for December)
By substituting these values into the linear function, we can solve for 'c' and 'd':
32.6 = c(6) + d, which simplifies to 6c + d = 32.6
32.4 = c(7) + d, which simplifies to 7c + d = 32.4
30.3 = c(8) + d, which simplifies to 8c + d = 30.3
26.4 = c(9) + d, which simplifies to 9c + d = 26.4
21.3 = c(10) + d, which simplifies to 10c + d = 21.3
18.0 = c(11) + d, which simplifies to 11c + d = 18.0
We can use a system of equations to solve for 'c' and 'd'. Solving this system will give us the values of 'c' and 'd'.
Once you have the values of 'a', 'b', 'c', and 'd', you can combine the two temperature functions to get the overall temperature function for all months, T(t).
Note: In this specific case, since we have a small number of data points and a clear linear pattern, using a linear function is reasonable. In more complex scenarios or with larger data sets, other mathematical models might be more appropriate.