Question

The temperature and time t given in hours from 0 to 24 after midnight in downtown mathville is given by t=10-5 sin(pi/12 t) degrees celcius. what is the average temperature between noon and midnight?

Answer 1

`T_A_v_g=9.918192559$^(\circ)C`

Step-by-step explanation:

The problem tell us that the temperature as function of time in downtown mathville is given by:

`T(t)=10-5*sin((\pi)/(12t))`

The average temperature over a given interval can be calculated as:

`T_a_v_g=(T_o+T_f)/(2)`

Where:

`T_o=Initial\hspace{3}temperature\\T_f=Final\hspace{3}temperature`

So, the initial temperature in this case, would be the temperature at noon, and the final temperature would be the temperature at midnight:

Therefore:

`T_o=T(12)=10-5*sin((\pi)/(12*12)) =9.890925575$^(\circ)C`

`T_f=T(24)=10-5*sin((\pi)/(12*24)) =9.945459543$^(\circ)C`

Hence, the average temperature between noon and midnight is:

`T_A_v_g=(9.890925575+9.945459543)/(2)=9.918192559$^(\circ)C`