Dry air is moving upward. If the ground temperature is 20circ and the | Mathematics I 2079

Mathematics I20795 marksNumerical: linear temperature modelLinear mathematical model

Dry air is moving upward. If the ground temperature is $20^\circ$ and the temperature at a height of 2km is $10^\circ$ , express the temperature $T$ in $^\circ$ C as a function of the height $h$ (in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km? [5]

Dry air is moving upward. If the ground temperature is $20^\circ$ and the temperature at a height of 2km is $10^\circ$, express the temperature $T$ in $^\circ$C as a function of the height $h$ (in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km? [5]

At ground level: $h = 0$ km, $T = 20^\circ$ C → point $(0, 20)$ - At height $h = 2$ km, $T = 10^\circ$ C → point $(2, 10)$ - Linear model assumed. --- Linear form: $T = mh + b$ . Slope: $m = \frac{T2 - T1}{h2 - h1} = \frac{10 - 20}{2 - 0} = \frac{-10}{2} = -5$ Intercept: at $h=0$ , $T=20$ , so

At ground level: $h = 0$ km, $T = 20^\circ$C → point $(0, 20)$ - At height $h = 2$ km, $T = 10^\circ$C → point $(2, 10)$ - Linear model assumed. --- Linear form: $T = mh + b$. Slope: $$m = \frac{T2 - T1}{h2 - h1} = \frac{10 - 20}{2 - 0} = \frac{-10}{2} = -5$$ Intercept: at $h=0$, $T=20$, so

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