Temperature data Hourly temperature data for Boulder, CO, San Francisco, CA, Nantucket, MA, and Duluth, MN, over a 12-hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The average temperature over the 12-hr period is .

Find an accurate approximation to the average temperature over the 12-hour period for San Francisco. State your method.

Problem 28E

Temperature data Hourly temperature data for Boulder, CO, San Francisco, CA, Nantucket, MA, and Duluth, MN, over a 12-hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function T(t). The averagetemperature over the 12-hr period is .

Find an accurate approximation to the average temperature over the 12-hour period for San Francisco. State your method.

Answer;

Step-1;

Given that temperature data hourly temperature data for Boulder , Co,San francisco ,CA, Nantucket ,MA , and Duluth ,MN over a 12 -hr period on the same day of January are shown in the figure.

And also given that these data from a continuous temperature function T(t) .The average temperature over the 12-hr period is = T(t) dt.

Given table is ;

Step-2 ;

Now , we have to find out an accurate approximation to the average temperature over the 12-hr period for San Francisco.

For our convenience in this problem we have to choose midpoint Rule approximations to the given integral.

Midpoint sum (also Midpoint approximation ) uses midpoint of subintervals;

f(x) dx (f() + f() +f() +.............+f()+f())

Where = , a = and b = , [a ,b] is the interval .

[ ] , [ ] , [ ] ……….. [ ] are the subintervals in [ a , b] .

Step-3;

Now , the given integral is ; = T(t) dt.

Now, we have to indicated midpoint Rule approximations to the given integral ,using n = 6 subintervals.

n= 6 , a = = 0 and b = = 12 , then = = = 2.

Divide interval [0, 12] into n =6 , subintervals of length = 2 with the following endpoints a = =...