Mathematics I208010 marksNumerical: linear model temperature functionLinear mathematical model
As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometer), assuming that linear model is appropriate. (a) Draw a graph of the function in part (b). Wha
As dry air moves upward, it expands and cools. If the ground temperature is $20°C$ and the temperature at height of 1 km is $10°C$, express the temperature $T$ (in $°C$) as a function of the height $h$ (in kilometer), assuming that linear model is appropriate.
(a) Draw a graph of the function in part (b). What does the slope represent?
(c) What is the temperature at a height of 2.5 km?
[5+5]
- Ground level: $h = 0$ km, $T = 20^{\circ}C$ → point $(0, 20)$ - At $h = 1$ km: $T = 10^{\circ}C$ → point $(1, 10)$ - Linear model assumed: $T = mh + b$ - Required: temperature at $h = 2.5$ km Slope: $$m = \frac{10 - 20}{1 - 0} = \frac{-10}{1} = -10$$ Intercept (from point $(0,20)$): $$b = 20$$ ...