Sure, let's solve the problem step by step:
1. Identify the given values:
- Initial temperature ([tex]\(T_{\text{initial}}\)[/tex]) = [tex]\(21^{\circ} C\)[/tex]
- Final temperature ([tex]\(T_{\text{final}}\)[/tex]) = [tex]\(75^{\circ} C\)[/tex]
- Heating time = 12 minutes
2. Determine the temperature change:
The change in temperature is calculated as:
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}} = 75 - 21 = 54^{\circ} C
\][/tex]
3. Calculate the rate of temperature change per minute:
Since the heating is happening at a constant rate, the rate of change of temperature per minute can be determined by dividing the temperature change by the time:
[tex]\[
\text{Rate of change} = \frac{\Delta T}{\text{Heating time}} = \frac{54}{12} = 4.5 \text{ degrees per minute}
\][/tex]
4. Formulate the temperature function [tex]\(T(x)\)[/tex]:
The temperature at any time [tex]\(x\)[/tex] in minutes can be expressed as:
[tex]\[
T(x) = T_{\text{initial}} + (\text{Rate of change}) \times x
\][/tex]
By substituting the known values:
[tex]\[
T(x) = 21 + 4.5x
\][/tex]
Therefore, the function's formula is:
[tex]\[
T = 21 + 4.5x
\][/tex]
