At 7 p. m. the temperature is 72 degrees f at 10 p. m. it is 52 degrees find the average rate of change in temperature

We will assume that we have a function relating between the time and the temperature. Any point in this function can be expressed as (x,y) where x is the time and y is the temperature (any point is expressed as (time, temperature)).

We have two points given:
At 7 p.m. the temperature is 72 degrees F: First point is (7,72)
At 10 p.m. the temperature is 52 degrees F: Second point is (10,52)

The rate of change is simply the slope of this function, this slope can be calculated as:
rate of change = (y2-y1) / (x2-x2)
we have y2= 52, y1= 72, x2= 10 and x1= 7

Substituting in the equation, we can calculate the rate of change as follows:
rate of change = (52-72) / (10-7) = -6.667 degrees/hours
Note: The negative sign indicated a DECREASE in the temperature.