To determine at what time the temperature will reach [tex]$100^\circ C$[/tex], we can use the line of best fit for the given data. Here is a step-by-step explanation:
1. Gather Data: The recorded times (in minutes) and corresponding temperatures (in degrees Celsius) are:
- Time (minutes): [tex]\([0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4, 4.5]\)[/tex]
- Temperature (°C): [tex]\([75, 79, 83, 86, 89, 91, 93, 94, 95, 95.5]\)[/tex]
2. Calculate the Line of Best Fit: The line of best fit for this data takes the form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the temperature
- [tex]\( x \)[/tex] is the time
- [tex]\( m \)[/tex] is the slope of the line
- [tex]\( b \)[/tex] is the y-intercept
3. Find the Slope and Intercept: Through analysis of the dataset:
- The slope ([tex]\(m\)[/tex]) of the line of best fit is approximately [tex]\(4.539\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is approximately [tex]\(77.836\)[/tex].
4. Formulate the Line of Best Fit Equation:
[tex]\[
\text{Temperature} = 4.539 \cdot \text{Time} + 77.836
\][/tex]
5. Determine the Time for [tex]$100^\circ C$[/tex]: We need to find the time ([tex]\(t\)[/tex]) when the temperature ([tex]\(T\)[/tex]) reaches [tex]\(100^\circ C\)[/tex]. Substitute [tex]\(T = 100\)[/tex] into the line of best fit equation:
[tex]\[
100 = 4.539 \cdot t + 77.836
\][/tex]
6. Solve for [tex]\(t\)[/tex]:
[tex]\[
100 = 4.539 \cdot t + 77.836
\][/tex]
[tex]\[
100 - 77.836 = 4.539 \cdot t
\][/tex]
[tex]\[
22.164 = 4.539 \cdot t
\][/tex]
[tex]\[
t = \frac{22.164}{4.539}
\][/tex]
[tex]\[
t \approx 4.883 \, \text{minutes}
\][/tex]
Therefore, the temperature will reach [tex]\(100^\circ C\)[/tex] at approximately [tex]\(4.883\)[/tex] minutes. Interpreting this in terms of the given multiple choice answers:
- The time is approximately 4.88 minutes, which is closest to 5 minutes.
So, according to the line of best fit, the time when the temperature will reach [tex]\(100^\circ C\)[/tex] is 5 minutes.
