Question

the weather report reveals that the temperature was 58 degrees at 9am and was 54 degrees at 9pm.
a. do you think the temperature decreased at a constant rate from 9am to 9pm? expla why or why not.
b. what is the average rate of change in temperature from 9am to 9pm? round to the nearest hundredth.
c. what information would you need to know to be able to determine about how fast temperature was changing at 2pm?

Explanation:

Step1: Analyze if temperature decreased at a constant rate

For a constant rate of change, the temperature should change uniformly over time. But in real - life, weather (temperature) is affected by many factors like sunlight intensity (which changes non - linearly throughout the day), cloud cover, etc. So, we cannot assume the temperature decreased at a constant rate from 9 am to 9 pm.

Step2: Calculate the average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
Let \(x_1\) be 9 am (we can consider \(x_1 = 9\) if we think of \(x\) as hours since midnight), \(x_2=21\) (9 pm is 21 hours since midnight), \(f(x_1) = 58\), and \(f(x_2)=54\).
The average rate of change \(=\frac{54 - 58}{21 - 9}=\frac{-4}{12}\approx - 0.33\) degrees per hour.

Step3: Determine information for 2 pm

To find the rate of change of temperature at 2 pm (14 hours since midnight), we would need to know the temperature values at times very close to 2 pm (for example, the temperature at 1:59 pm and 2:01 pm) so that we can use the formula \(\lim_{h
ightarrow0}\frac{f(14 + h)-f(14 - h)}{(14 + h)-(14 - h)}\) (the definition of the derivative, which gives the instantaneous rate of change). In a more practical sense, we would need a function that models the temperature over time (either from a set of data points that are very close to 2 pm or a known temperature - time function)

Answer:

a. No. Weather factors like sunlight and cloud cover cause non - uniform temperature change.
b. \(-0.33\) degrees per hour.
c. Temperature values at times very close to 2 pm (e.g., 1:59 pm and 2:01 pm) or a temperature - time function.