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

Question

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

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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:  T(t) = 24.5 - 5.5 sin((2π(t+1))/24).  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:  -5.5 sin((2π(t+1))/24) = -5.5 * (-1).  This simplifies to:  -5.5 sin((2π(t+1))/24) = 5.5.Solve for t: Now we need to solve for t when sin((2π(t+1))/24) = -1. The general solution for sin(θ) = -1 is θ = 3π/2 + 2kπ, where k is an integer.  So we have:  (2π(t+1))/24 = 3π/2 + 2kπ.  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:  2π(t+1) = 36π + 48kπ.  Next, we divide both sides by 2π:  t+1 = 18 + 24k.Find Specific Time: Now we subtract 1 from both sides to solve for t:  t = 17 + 24k.  Since we are looking for the time within the first 24-hour period after midnight, we take k = 0:  t = 17 hours after midnight.

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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