Question

Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about$1^{\circ }C$for each 100-m rise, up to about 12km.

1. If the ground temperature is${20}^{\circ }C$, write a formula for the temperature at height h.
2. What range of temperature can be expected if a plane takes off and reaches a maximum height of 5 km?

Short answer

1. The Formula for the temperature at the height h,if the ground temperature is ${20}^{\circ }C$is $T_h=20-\frac{h}{100}$.
2. The range of temperature if a plane takes off and reaches a maximum height of 5 km is $-30\le T_h\le 20$.

Step-by-step solution

Part a. Step 1. Apply the concept of mathematical model.

A mathematical model is an equation that describes a real world object or process. Modeling is a process of finding such equations.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Part a. Step 2. Find the Formula.

Let$T_h$ be the air temperature at a height h above the ground level. Since the temperature reduces every 100m above the ground level, the following equation is obtained:

$T_h=20-\frac{h}{100},h\le 12000m$

Part a. Step 3. Example using the Formula.

If $h=200m$and the air temperature is given by

$$\begin{gathered} T_h=20-\frac{200}{100} \\ =20-2 \\ ={18}^{\circ }C \end{gathered}$$

Part b. Step 1. Apply the concept of mathematical model.

A mathematical model is an equation that describes a real world object or process. Modeling is a process of finding such equations.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Part b. Step 2. Solve the equation.

In order to find the range of temperature, solve the equation obtained in step 2 of a.

$$\begin{gathered} T_h=20-\frac{h}{100} \\ \Rightarrow T_h-20=-\frac{h}{100}\text{(subtract20)} \\ \Rightarrow 20-T_h=\frac{h}{100}\text{(multiplybynegativesign)} \\ \Rightarrow h=2000-100T_h\text{(multiplyby100)} \end{gathered}$$

Since the range of temperature is to be found when the plane takes off and reaches a maximum height of $5km(=5000m),$use the inequality $0\le h\le 5000$.

Hence,

$$\begin{gathered} 0\le h\le 5000 \\ \Rightarrow 0\le 2000-100T_h\le 5000 \\ \Rightarrow 0-2000\le -100T_h\le 5000-2000\text{(subtract2000)} \\ \Rightarrow -2000\le -100T_h\le 3000 \\ \Rightarrow 2000\ge 100T_h\ge -3000 \end{gathered}$$

Part b. Step 3. Find the range

Isolate the variable from the inequality $2000\ge 100T_h\ge -3000$.

Hence, divide the inequality by 100. Therefore,

$\begin{array}{c}\frac{2000}{100}\ge \frac{100T_h}{100}\ge \frac{-3000}{100}\\ 20\ge T_h\ge -30\\ -30\le T_h\le 20\end{array}$

is the required range of temperature.