In San Diego, the number of daylight hours is dependent on the day of the year. The number of daylight hours $D$ can be modeled by the sinusoidal function $D(t)=2.4 \sin (0.017(t-81))+12$ , where $t$ is the day of the year from $t=1$ to $t=365$ . Based on this model, which of the following statements is correct? $\newline$ (A) The minimum number of daylight hours in San Diego is $12$ hours. $\newline$ (B) The maximum number of daylight hours in San Diego is $12$ hours. $\newline$ (C) The minimum number of daylight hours in San Diego is $9$ . $6$ hours. $\newline$ (D) The maximum number of daylight hours in San Diego is $9$ . $6$ hours.

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Q. In San Diego, the number of daylight hours is dependent on the day of the year. The number of daylight hours

D

can be modeled by the sinusoidal function

D(t)=2.4 \sin (0.017(t-81))+12

, where

t

is the day of the year from

t=1

t=365

. Based on this model, which of the following statements is correct?

\newline

(A) The minimum number of daylight hours in San Diego is

12

hours.

\newline

(B) The maximum number of daylight hours in San Diego is

12

hours.

\newline

9

6

hours.

\newline

(D) The maximum number of daylight hours in San Diego is

9

6

hours.

Understand Sine Function: The sinusoidal function provided is $D(t)=2.4 \sin(0.017(t-81))+12$ . To find the minimum and maximum number of daylight hours, we need to understand the properties of the sine function. The sine function oscillates between $-1$ and $1$ , so the amplitude of the sine function in this case is $2.4$ . This means the function $D(t)$ will oscillate $2.4$ units above and below its midline, which is at $12$ hours.
Find Maximum Daylight Hours: To find the maximum number of daylight hours, we add the amplitude to the midline value: Maximum daylight hours = Midline + Amplitude = $12 + 2.4 = 14.4$ hours.
Find Minimum Daylight Hours: To find the minimum number of daylight hours, we subtract the amplitude from the midline value: Minimum daylight hours = Midline $-$ Amplitude $= 12 - 2.4 = 9.6$ hours.
Compare Findings with Given Statements: Now we can compare our findings with the given statements. The minimum number of daylight hours is $9.6$ hours, which makes option (C) correct. The maximum number of daylight hours is $14.4$ hours, which is not listed in the options, but it does tell us that option (B) is incorrect since it states the maximum is $12$ hours.
Identify Correct Statement: Since the maximum number of daylight hours is $14.4$ , which is not one of the options, we can conclude that the correct statement regarding the minimum number of daylight hours is the only one we need to identify. Therefore, the correct statement is (C) The minimum number of daylight hours in San Diego is $9.6$ hours.