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Sagot :
To determine whether the temperature in degrees Fahrenheit is proportional to the temperature in degrees Celsius, we need to analyze the given data and graph the relationship on the coordinate plane.
### Step-by-Step Solution:
1. Understanding Proportional Relationships:
- For two quantities to be proportional, the ratio between them must be constant. This means if we have two quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex], they are proportional if [tex]\( \frac{y}{x} \)[/tex] is the same for all values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. Analyze the Given Data:
- We are provided with temperatures in degrees Celsius (C) and their corresponding temperatures in degrees Fahrenheit (F):
[tex]\[ \begin{array}{|c|c|} \hline \text{Celsius (C)} & \text{Fahrenheit (F)} \\ \hline 0 & 32 \\ 5 & 41 \\ 10 & 50 \\ 15 & 59 \\ 20 & 68 \\ \hline \end{array} \][/tex]
3. Graph the Data:
- Plot the points [tex]\((0, 32)\)[/tex], [tex]\((5, 41)\)[/tex], [tex]\((10, 50)\)[/tex], [tex]\((15, 59)\)[/tex], and [tex]\((20, 68)\)[/tex] on the coordinate plane where the x-axis represents Celsius and the y-axis represents Fahrenheit.
- Draw a line through the points.
4. Examine the Line:
- If the points form a straight line that goes through the origin (0,0), then F and C are proportional. Otherwise, they are not.
5. Calculate the Ratios:
- Calculate the ratios [tex]\( \frac{F - 32}{C} \)[/tex] to see if they are constant:
[tex]\[ \begin{array}{c|c} \text{Celsius (C)} & \text{Ratio} \\ \hline 5 & \frac{41 - 32}{5} = \frac{9}{5} = 1.8 \\ 10 & \frac{50 - 32}{10} = \frac{18}{10} = 1.8 \\ 15 & \frac{59 - 32}{15} = \frac{27}{15} = 1.8 \\ 20 & \frac{68 - 32}{20} = \frac{36}{20} = 1.8 \\ \end{array} \][/tex]
The ratios are consistent: 1.8 for each pair of [tex]\( C \)[/tex] and [tex]\( F \)[/tex].
### Conclusion:
The temperature in degrees Fahrenheit (F) is proportional to the temperature in degrees Celsius (C) with a constant ratio of 1.8 once we account for the offset of 32 degrees Fahrenheit at 0 degrees Celsius. This means for each degree Celsius, the Fahrenheit increases by 1.8 degrees once we offset the freezing point of water. This relationship can be expressed by the equation [tex]\( F = 1.8C + 32 \)[/tex].
So, the temperature in degrees Fahrenheit is indeed proportional to the temperature in degrees Celsius, considering the slope (1.8). Thus, graphing this relationship will show a straight line but not through the origin because of the temperature offset in the Fahrenheit scale at 0 degrees Celsius.
### Step-by-Step Solution:
1. Understanding Proportional Relationships:
- For two quantities to be proportional, the ratio between them must be constant. This means if we have two quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex], they are proportional if [tex]\( \frac{y}{x} \)[/tex] is the same for all values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. Analyze the Given Data:
- We are provided with temperatures in degrees Celsius (C) and their corresponding temperatures in degrees Fahrenheit (F):
[tex]\[ \begin{array}{|c|c|} \hline \text{Celsius (C)} & \text{Fahrenheit (F)} \\ \hline 0 & 32 \\ 5 & 41 \\ 10 & 50 \\ 15 & 59 \\ 20 & 68 \\ \hline \end{array} \][/tex]
3. Graph the Data:
- Plot the points [tex]\((0, 32)\)[/tex], [tex]\((5, 41)\)[/tex], [tex]\((10, 50)\)[/tex], [tex]\((15, 59)\)[/tex], and [tex]\((20, 68)\)[/tex] on the coordinate plane where the x-axis represents Celsius and the y-axis represents Fahrenheit.
- Draw a line through the points.
4. Examine the Line:
- If the points form a straight line that goes through the origin (0,0), then F and C are proportional. Otherwise, they are not.
5. Calculate the Ratios:
- Calculate the ratios [tex]\( \frac{F - 32}{C} \)[/tex] to see if they are constant:
[tex]\[ \begin{array}{c|c} \text{Celsius (C)} & \text{Ratio} \\ \hline 5 & \frac{41 - 32}{5} = \frac{9}{5} = 1.8 \\ 10 & \frac{50 - 32}{10} = \frac{18}{10} = 1.8 \\ 15 & \frac{59 - 32}{15} = \frac{27}{15} = 1.8 \\ 20 & \frac{68 - 32}{20} = \frac{36}{20} = 1.8 \\ \end{array} \][/tex]
The ratios are consistent: 1.8 for each pair of [tex]\( C \)[/tex] and [tex]\( F \)[/tex].
### Conclusion:
The temperature in degrees Fahrenheit (F) is proportional to the temperature in degrees Celsius (C) with a constant ratio of 1.8 once we account for the offset of 32 degrees Fahrenheit at 0 degrees Celsius. This means for each degree Celsius, the Fahrenheit increases by 1.8 degrees once we offset the freezing point of water. This relationship can be expressed by the equation [tex]\( F = 1.8C + 32 \)[/tex].
So, the temperature in degrees Fahrenheit is indeed proportional to the temperature in degrees Celsius, considering the slope (1.8). Thus, graphing this relationship will show a straight line but not through the origin because of the temperature offset in the Fahrenheit scale at 0 degrees Celsius.
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