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Answer :
Sure! Let's tackle this problem step-by-step, focusing on each part separately.
### Part a: Rate of Change
We need to find the rate of change of the Fahrenheit temperature concerning each unit change in Celsius temperature. Since the Fahrenheit temperature [tex]\( F(C) \)[/tex] is expressed as a linear function of the Celsius temperature [tex]\( C \)[/tex], we have:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
In this linear function, the coefficient of [tex]\( C \)[/tex] is 1.8. This means that for every 1-degree change in Celsius, the Fahrenheit temperature changes by 1.8 degrees. Therefore, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
### Part b: Find and Interpret [tex]\( F(25) \)[/tex]
Now, we want to find the Fahrenheit temperature when the Celsius temperature is 25 degrees. We'll use the function:
[tex]\[ F(C) = 1.8 \times C + 32 \][/tex]
Substitute [tex]\( C = 25 \)[/tex]:
[tex]\[ F(25) = 1.8 \times 25 + 32 \][/tex]
[tex]\[ F(25) = 45 + 32 \][/tex]
[tex]\[ F(25) = 77 \][/tex]
So, at 25 degrees Celsius, the temperature is 77 degrees Fahrenheit. The interpretation is: At 25 degrees Celsius, it is 77.0 degrees Fahrenheit.
### Part c: Calculate [tex]\( F(-50) \)[/tex]
Finally, we find the Fahrenheit temperature when the Celsius temperature is [tex]\(-50\)[/tex] degrees. Using the same function:
[tex]\[ F(C) = 1.8 \times C + 32 \][/tex]
Substitute [tex]\( C = -50 \)[/tex]:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 \][/tex]
[tex]\[ F(-50) = -58 \][/tex]
Thus, at [tex]\(-50\)[/tex] degrees Celsius, the temperature is [tex]\(-58\)[/tex] degrees Fahrenheit.
Overall, we've found the rate of change, computed and interpreted [tex]\( F(25) \)[/tex], and calculated [tex]\( F(-50) \)[/tex] using the linear function for temperature conversion.
### Part a: Rate of Change
We need to find the rate of change of the Fahrenheit temperature concerning each unit change in Celsius temperature. Since the Fahrenheit temperature [tex]\( F(C) \)[/tex] is expressed as a linear function of the Celsius temperature [tex]\( C \)[/tex], we have:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
In this linear function, the coefficient of [tex]\( C \)[/tex] is 1.8. This means that for every 1-degree change in Celsius, the Fahrenheit temperature changes by 1.8 degrees. Therefore, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
### Part b: Find and Interpret [tex]\( F(25) \)[/tex]
Now, we want to find the Fahrenheit temperature when the Celsius temperature is 25 degrees. We'll use the function:
[tex]\[ F(C) = 1.8 \times C + 32 \][/tex]
Substitute [tex]\( C = 25 \)[/tex]:
[tex]\[ F(25) = 1.8 \times 25 + 32 \][/tex]
[tex]\[ F(25) = 45 + 32 \][/tex]
[tex]\[ F(25) = 77 \][/tex]
So, at 25 degrees Celsius, the temperature is 77 degrees Fahrenheit. The interpretation is: At 25 degrees Celsius, it is 77.0 degrees Fahrenheit.
### Part c: Calculate [tex]\( F(-50) \)[/tex]
Finally, we find the Fahrenheit temperature when the Celsius temperature is [tex]\(-50\)[/tex] degrees. Using the same function:
[tex]\[ F(C) = 1.8 \times C + 32 \][/tex]
Substitute [tex]\( C = -50 \)[/tex]:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 \][/tex]
[tex]\[ F(-50) = -58 \][/tex]
Thus, at [tex]\(-50\)[/tex] degrees Celsius, the temperature is [tex]\(-58\)[/tex] degrees Fahrenheit.
Overall, we've found the rate of change, computed and interpreted [tex]\( F(25) \)[/tex], and calculated [tex]\( F(-50) \)[/tex] using the linear function for temperature conversion.
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