Question

use answer questions 14-15
the table shows the average monthly precipitation in inches, in san francisco, ca, based on data from 1961 to 1990.

month	jan	feb	mar	apr	may	june	july	aug	sept	oct	nov	dec

1. does the data appear to be periodic? does it make a wave pattern (down up down up)?

a. true
b. false

1. if so, what is the equation for the data?

a. $2.3 \sin(1.8x + 0.4)+2.5$
b. $2.5 \sin(0.4x + 1.8)+2.3$
c. $1.8 \sin(2.3x - 2.5)$

Explanation:

Step1: Analyze periodicity of data

Check the precipitation values: Jan(4.4) → Feb(3.2) → Mar(3.1) → Apr(1.4) → May(0.2) → Jun(0.1) → Jul(0) (decreasing trend); then Aug(0.1) → Sep(0.2) → Oct(1.2) → Nov(2.9) → Dec(3.1) (increasing trend). This forms a repeating wave-like pattern over 12 months, so it is periodic.

Step2: Identify sinusoidal equation parameters

The general sinusoidal form is $y = A\sin(Bx + C) + D$, where:

• $D$ = vertical shift (average of max/min values): $\frac{4.4 + 0}{2} = 2.2$, closest to 2.3
• $A$ = amplitude (half the range): $\frac{4.4 - 0}{2} = 2.2$, closest to 2.3
• $B$ = $\frac{2\pi}{12} \approx 0.52$, closest to 0.4

Match to the options: Option B has $A=2.5$, $B=0.4$, $D=2.3$, which aligns with the approximate calculated values.

Answer:

1. A. True
2. B. 2.5 sin (0.4x + 1.8) + 2.3