In a certain city the temperature (in degrees Fahrenheit) t hours after 9am was

Question

In a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function

Determine the temperature at 9 am.
Determine the temperature at 3 pm.
Find the average temperature during the period from 9 am to 9 pm.

In a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function T(t) = 70 + 15 sin left(frac{pi t}{12} right) Determine the temperature at 9 am. Determine the temperature at 3 pm. Find the average temperature during the period from 9 am to 9 pm.

Explanation / Answer

a) at 9 am, t = 0
T(0) = 70 + 15sin(0) = 70 degrees Fahrenheit Ans.

b) at 3 pm, t = 6 (hrs after 9 am)
T(6) = 70 + 15sin(6pi/12) = 70 +15sin(pi/2) = 70 +15 = 85 degrees Fahrenheit Ans.

c). If t is the number of hours after 9am, then 9am corresponds to t = 0 and 9pm corresponds to t = 12. The average temp over this time interval is

12
? T(t) dt / (12 - 0)
0

For the given T, this is

12
? 70 + 15 sin(? t/12) dt / 12
0

= (70t - (15*12/?) cos(? t/12))/12 evaluated from 0 to 12
= (70*12 - (180/?)cos(?))/12 - (0 - (180/?) cos(0))/12
= 70 + 15/? + 15/?
= 70 + 30/?
= 79.55

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