Part A
The change in elevation is 1,000 feet, which we can represent as the rational number \(1000\). The change in temperature is a drop of 20 degrees Fahrenheit, represented as the rational number \(-20\). Therefore:
- Change in elevation: \(1000\)
- Change in temperature: \(-20\)
Part B
The numerical expression to represent the rate of temperature change in degrees Fahrenheit per foot is:
\[
\frac{-20}{1000}
\]
Part C
To find the rate of temperature change in degrees Fahrenheit per foot, we simplify the expression from Part B:
\[
\frac{-20}{1000} = -0.02
\]
Thus, the rate of temperature change is \(-0.02\) degrees Fahrenheit per foot.
Part D
For the reverse situation, as the airplane descends, the change in elevation is still \(1,000\) feet (which we can represent as \(1000\)) but the temperature rises by \(20\) degrees Fahrenheit, so we represent that as \(20\). The numerical expression becomes:
\[
\frac{20}{1000}
\]
Part E
We find the rate of temperature change in degrees Fahrenheit per foot by simplifying the expression from Part D:
\[
\frac{20}{1000} = 0.02
\]
Thus, the rate of temperature change is \(0.02\) degrees Fahrenheit per foot.
Part F
The rate found in Part E (\(0.02\) degrees Fahrenheit per foot) is the opposite of the rate found in Part C (\(-0.02\) degrees Fahrenheit per foot). This means that the temperature increases as the airplane descends and decreases as it climbs, illustrating a direct relationship between elevation change and temperature change. When the elevation decreases, the temperature increases by the same rate, and vice versa.
Self-Evaluation
I rate my work as a 5. I was able to set up and solve each part correctly and understand the relationship between the two situations. This exercise helped me learn how to represent rates of change with rational numbers clearly. The calculations were straightforward, but I had to ensure I understood how to represent the temperature change as positive and negative values for ascent and descent. Overall, I found it challenging to remember to include the negatives, but the step-by-step breakdown helped clarify that process.
1 answer