a.

Estimate the temparature on January 31.

The average daily temperature in degrees Fahrenheit for a city can be modeled by , where is a function of time.

Since represents January 15, represents January 31.

Draw a coordinate plane..

Graph the function .

Graph :

Observe the above graph :

At the temparature is .

Therefore, the temperature on January 31 is about .

b.

Find the number of months that the average daily temperature is greater than throughout the entire month.

Let the function be .

Draw a coordinate plane.

Graph the two functions and in the same window.

Observe the graph :

The intersection ponts are and .

on the interval  .

Because corresponds to about the middle of April, the temperature does not exceed for the entire month until May. Therefore, the average daily temperature is greater than for the entire month during the months of May, June, and July.

Therefore, the average the temperature was higher than for three months.

c.

Graph of is : 

Observe the graph of the function :

The heighest temparature occurs at .

At , the temparature is .

From the given data represents January 15, represents February 15,........., represents June 15.

Therefore, the maximum average monthly temperature is during the month of June.

a. The temperature on January 31 is about .

b. The average daily temperature is greater than for the entire month during the months of May, June, and July.

c. The maximum average monthly temperature is during the month of June.