Sure, let's derive the formula step-by-step for the temperature, [tex]\( T \)[/tex], as a function of the heating time, [tex]\( x \)[/tex], in minutes.
1. Initial Temperature:
The initial temperature of the chemical solution is [tex]\( 21^{\circ} C \)[/tex].
2. Final Temperature:
After 12 minutes of heating, the temperature rises to [tex]\( 75^{\circ} C \)[/tex].
3. Change in Temperature:
The change in temperature ([tex]\(\Delta T\)[/tex]) over the 12 minutes is the final temperature minus the initial temperature:
[tex]\[
\Delta T = 75^{\circ} C - 21^{\circ} C = 54^{\circ} C
\][/tex]
4. Time Duration:
The time duration for the change in temperature is [tex]\( 12 \)[/tex] minutes.
5. Rate of Heating:
The rate of heating ([tex]\( r \)[/tex]) is the change in temperature divided by the time duration:
[tex]\[
r = \frac{\Delta T}{\text{time duration}} = \frac{54^{\circ} C}{12 \text{ minutes}} = 4.5^{\circ} C/\text{minute}
\][/tex]
6. Function Formula:
The temperature [tex]\( T \)[/tex] after [tex]\( x \)[/tex] minutes of heating can be found by adding the initial temperature to the product of the rate of heating and the time [tex]\( x \)[/tex]:
[tex]\[
T = \text{initial temperature} + (\text{rate of heating} \times x)
\][/tex]
Substituting the known values, we get:
[tex]\[
T = 21 + 4.5 \cdot x
\][/tex]
Thus, the formula for the temperature [tex]\( T \)[/tex] as a function of the heating time [tex]\( x \)[/tex] in minutes is:
[tex]\[
T = 21 + 4.5x
\][/tex]
