Question

22 mark for review
month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
temperature (degrees celsius) | 6.1 | -5.5 | -6.0 | 10.0 | 17.2 | 25.6 | 30.6 | 32.2 | 26.1
the table gives the maximum temperature, in degrees celsius, on the first day of each of nine months in a certain city. the function ( f ) given by ( f(\theta)=asin(b(\theta + c)) + d ), where ( a, b, c, ) and ( d ) are constants, is used to model these data with ( \theta ) representing the number of the month. assume that the period of ( f ) is 12 months. based on the data in the table, which of the following is the best value for ( d )?

Explanation:

Step1: Identify d's meaning

In the sinusoidal function $f(\theta)=a\sin(b(\theta+c))+d$, $d$ represents the vertical shift, which is the average of the maximum and minimum values of the function.

Step2: Find max and min temps

From the table, maximum temperature $T_{max}=32.2$, minimum temperature $T_{min}=-6.0$.

Step3: Calculate average (d)

Compute the average of max and min:
$$d=\frac{T_{max}+T_{min}}{2}=\frac{32.2 + (-6.0)}{2}$$
$$=\frac{26.2}{2}=13.1$$

Answer:

13.1