In order to find the temperature, we must find the intensity of the light at the time we are evaluating.
Since the function is model since 6:00 am and we need the temperature at 2:00 pm we need to know how many hours have passed from 6:00 am to 2:00 pm.
This is 8 hours.
Then,
[tex]t=8h[/tex]
Now that we have the corresponding time we can find the intensity by replacing t for 8.
[tex]\begin{gathered} I=\frac{12h-h^2}{36} \\ I=\frac{12\cdot8-8^2}{36} \\ I=\frac{96-64}{36} \\ I=\frac{32}{36} \\ I=\frac{8}{9} \end{gathered}[/tex]
Continue by calculating the temperature using the function for the temperature with the intensity of the light calculated at 2:00 pm
[tex]\begin{gathered} T=\sqrt[]{5000\cdot I} \\ T=\sqrt[]{5000\cdot\frac{8}{9}} \\ T=\sqrt[]{\frac{40000}{9}} \\ T=\frac{200}{3}\approx66.67\cong67 \end{gathered}[/tex]
The closest to the temperature is 67.
