Questions: In this example we considered the function W=f(T, v), where W is the wind-chill index, T is the actual temperature, and v is the wind speed. A numerical representation is given in this table. (a) What is the value of f(-15,60) ? What is its meaning? We get f(-15,60)= , which means that if the temperature is -15°C and the wind speed is 60 km / h, then the air would feel equivalent to approximately °C without wind. (b) Describe in words the meaning of the question "For what value of v is f(-20, v)=-30 ?" The question is asking: when the temperature is °C, what wind speed gives a wind-chill index of °C ? Answer the question. v= km / h (c) Describe in words the meaning of the question "For what value of T is f(T, 20)=-43 ?" The question is asking: when the wind speed is km / h, what temperature gives a wind-chill Index of °C ? Answer the question. T= °C (d) What is the meaning of the function W=f(-10, v) ? Describe the behavior of this function. The function W=f(-10, v) means that we fix T at and allow v to vary, resulting in a function of one variable. In other words, the function gives wind-chill index values for different wind speeds when the temperature is °C. From the table (look at the row corresponding to T=-10 ), the function - Select - and appears to approach a constant value as v increases. (e) What is the meaning of the function W=f(T, 80) ? Describe the behavior of this function. The function W=f(T, 80) means that we fix v at and allow T to vary, again giving a function of one variable. In other words, the function gives wind-chill index values for different temperatures when the wind speed is km / h. From the table (look at the column corresponding to v=80 ), the function -Select - almost linearly as T Increases.

In this example we considered the function W=f(T, v), where W is the wind-chill index, T is the actual temperature, and v is the wind speed. A numerical representation is given in this table. (a) What is the value of f(-15,60) ? What is its meaning? We get f(-15,60)= , which means that if the temperature is -15°C and the wind speed is 60 km / h, then the air would feel equivalent to approximately °C without wind. (b) Describe in words the meaning of the question "For what value of v is f(-20, v)=-30 ?" The question is asking: when the temperature is °C, what wind speed gives a wind-chill index of °C ? Answer the question. v= km / h (c) Describe in words the meaning of the question "For what value of T is f(T, 20)=-43 ?" The question is asking: when the wind speed is km / h, what temperature gives a wind-chill Index of °C ? Answer the question. T= °C (d) What is the meaning of the function W=f(-10, v) ? Describe the behavior of this function. The function W=f(-10, v) means that we fix T at and allow v to vary, resulting in a function of one variable. In other words, the function gives wind-chill index values for different wind speeds when the temperature is °C. From the table (look at the row corresponding to T=-10 ), the function - Select - and appears to approach a constant value as v increases. (e) What is the meaning of the function W=f(T, 80) ? Describe the behavior of this function. The function W=f(T, 80) means that we fix v at and allow T to vary, again giving a function of one variable. In other words, the function gives wind-chill index values for different temperatures when the wind speed is km / h. From the table (look at the column corresponding to v=80 ), the function -Select - almost linearly as T Increases.
Transcript text: In this example we considered the function $W=f(T, v)$, where $W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is the wind speed. A numerical representation is given in this table. (a) What is the value of $f(-15,60)$ ? What is its meaning? We get $f(-15,60)=$ $\square$ , which means that if the temperature is $-15^{\circ} \mathrm{C}$ and the wind speed is $60 \mathrm{~km} / \mathrm{h}$, then the air would feel equivalent to approximately $\square$ ${ }^{\circ} \mathrm{C}$ without wind. (b) Describe in words the meaning of the question "For what value of $v$ is $f(-20, v)=-30$ ?" The question is asking: when the temperature is $\square$ ${ }^{\circ} \mathrm{C}$, what wind speed gives a wind-chill index of $\square$ ${ }^{\circ} \mathrm{C}$ ? Answer the question. $v=$ $\square$ $\mathrm{km} / \mathrm{h}$ (c) Describe in words the meaning of the question "For what value of $T$ is $f(T, 20)=-43$ ?" The question is asking: when the wind speed is $\square$ $\mathrm{km} / \mathrm{h}$, what temperature gives a wind-chill Index of $\square$ ${ }^{\circ} \mathrm{C}$ ? Answer the question. $T=$ $\square$ ${ }^{\circ} \mathrm{C}$ (d) What is the meaning of the function $W=f(-10, v)$ ? Describe the behavior of this function. The function $W=f(-10, v)$ means that we fix $T$ at $\square$ and allow $v$ to vary, resulting in a function of one variable. In other words, the function gives wind-chill index values for different wind speeds when the temperature is $\qquad$ ${ }^{\circ} \mathrm{C}$. From the table (look at the row corresponding to $T=-10$ ), the function - Select $-\checkmark$ and appears to approach a constant value as $v$ increases. (e) What is the meaning of the function $W=f(T, 80)$ ? Describe the behavior of this function. The function $W=f(T, 80)$ means that we fix $v$ at $\square$ and allow $T$ to vary, again giving a function of one variable. In other words, the function gives wind-chill index values for different temperatures when the wind speed is $\square$ $\mathrm{km} / \mathrm{h}$. From the table (look at the column corresponding to $v=80$ ), the function -Select $-v$ almost linearly as $T$ Increases.
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Solution

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Solution Steps

Solution Approach

(a) To find the value of \( f(-15, 60) \), we need to look up the value in the given table where the temperature \( T \) is \(-15\) and the wind speed \( v \) is \(60\). This value represents the wind-chill index, which indicates how cold it feels at that temperature and wind speed.

(b) The question asks for the wind speed \( v \) that results in a wind-chill index of \(-30\) when the temperature is \(-20\). We need to find the value of \( v \) in the table where \( f(-20, v) = -30 \).

(c) The question asks for the temperature \( T \) that results in a wind-chill index of \(-43\) when the wind speed is \(20\). We need to find the value of \( T \) in the table where \( f(T, 20) = -43 \).

Step 1: Determine \( f(-15, 60) \)

To find the value of \( f(-15, 60) \), we look up the table where the temperature \( T = -15 \) and the wind speed \( v = 60 \). The table provides the wind-chill index for these conditions.

Step 2: Interpret the Meaning of \( f(-15, 60) \)

The value \( f(-15, 60) = -30 \) means that when the actual temperature is \(-15^\circ \mathrm{C}\) and the wind speed is \(60 \, \mathrm{km/h}\), the air feels like \(-30^\circ \mathrm{C}\) without wind.

Step 3: Find \( v \) for \( f(-20, v) = -30 \)

We need to find the wind speed \( v \) such that \( f(-20, v) = -30 \). From the table, when \( T = -20 \), the wind speed \( v = 60 \) results in a wind-chill index of \(-30\).

Step 4: Interpret the Meaning of \( f(-20, v) = -30 \)

This means that when the temperature is \(-20^\circ \mathrm{C}\), a wind speed of \(60 \, \mathrm{km/h}\) makes the air feel like \(-30^\circ \mathrm{C}\).

Step 5: Find \( T \) for \( f(T, 20) = -43 \)

We need to find the temperature \( T \) such that \( f(T, 20) = -43 \). However, the table does not provide a value for \( T \) that satisfies this condition, as indicated by the result \( T = \text{None} \).

Final Answer

  • \( f(-15, 60) = \boxed{-30} \)
  • \( v \) for \( f(-20, v) = -30 \) is \(\boxed{60 \, \mathrm{km/h}}\)
  • \( T \) for \( f(T, 20) = -43 \) is \(\boxed{\text{None}}\) (no such temperature exists in the table)
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