In January, the average temperature $t$ hours after midnight in Mumbai, India, is given by $\newline$ $T(t)=24.5-5.5 \sin \left(\frac{2 \pi(t+1)}{24}\right) .$ $\newline$ What is the coldest time of day in Mumbai? Give an exact answer. $\newline$ $\square$ hours after midnight

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Convert between Celsius and Fahrenheit

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Q. In January, the average temperature

t

hours after midnight in Mumbai, India, is given by

\newline

T(t)=24.5-5.5 \sin \left(\frac{2 \pi(t+1)}{24}\right) .

\newline

What is the coldest time of day in Mumbai? Give an exact answer.

\newline

\square

hours after midnight

Determine Minimum Temperature: To find the coldest time of day, we need to determine when the temperature function $T(t)$ is at its minimum. The temperature function is given by: $\newline$ $T(t) = 24.5 - 5.5 \sin\left(\frac{2\pi(t+1)}{24}\right)$ . $\newline$ The minimum temperature occurs when the sine function reaches its minimum value, which is $-1$ .
Set Sine Function: Since the sine function varies between $-1$ and $1$ , we set the sine function to $-1$ to find the time when the temperature is at its minimum: $\newline$ $-5.5 \sin\left(\frac{2\pi(t+1)}{24}\right) = -5.5 \times (-1)$ . $\newline$ This simplifies to: $\newline$ $-5.5 \sin\left(\frac{2\pi(t+1)}{24}\right) = 5.5$ .
Solve for $t$ : Now we need to solve for $t$ when $\sin\left(\frac{2\pi(t+1)}{24}\right) = -1$ . The general solution for $\sin(\theta) = -1$ is $\theta = \frac{3\pi}{2} + 2k\pi$ , where $k$ is an integer. $\newline$ So we have: $\newline$ $\frac{2\pi(t+1)}{24} = \frac{3\pi}{2} + 2k\pi$ . $\newline$ We will solve for $t$ to find the specific time after midnight.
Multiply and Divide: First, we multiply both sides by $24$ to get rid of the denominator: $\newline$ $2\pi(t+1) = 36\pi + 48k\pi$ . $\newline$ Next, we divide both sides by $2\pi$ : $\newline$ $t+1 = 18 + 24k$ .
Find Specific Time: Now we subtract $1$ from both sides to solve for $t$ : $\newline$ $t = 17 + 24k$ . $\newline$ Since we are looking for the time within the first $24$ -hour period after midnight, we take $k = 0$ : $\newline$ $t = 17$ hours after midnight.