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The temperature of a chemical solution is originally [tex]$21^{\circ} C$[/tex]. A chemist heats the solution at a constant rate, and the temperature of the solution is [tex]$75^{\circ} C$[/tex] after 12 minutes of heating.

The temperature, [tex][tex]$T$[/tex][/tex], of the solution in [tex]^{\circ} C[/tex] is a function of [tex]$x$[/tex], the heating time in minutes.

Write the function's formula:
[tex]\[ T = \square \][/tex]



Answer :

GinnyAnswer
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]

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