The temperature $\mathrm{T}\left({ }^{\circ} \mathrm{F}\right)$ of a room at time $\mathrm{t}$ minutes is given by the formula below. \[ T=65-12 \sqrt{25-t} \text { for } 0 \leq t \leq 25 \] a. Find the room's temperature when $t=0, t=16$, and $t=25$. b. Find the room's average temperature for $0 \leq t \leq 25$.

Step 1 ：Given the temperature function \(T=65-12 \sqrt{25-t}\) for \(0 \leq t \leq 25\)

Step 2 ：Substitute \(t=0\) into the function to find the temperature at \(t=0\), which gives \(T_0 = 65-12 \sqrt{25-0} = 5.0\) degrees Fahrenheit

Step 3 ：Substitute \(t=16\) into the function to find the temperature at \(t=16\), which gives \(T_{16} = 65-12 \sqrt{25-16} = 29.0\) degrees Fahrenheit

Step 4 ：Substitute \(t=25\) into the function to find the temperature at \(t=25\), which gives \(T_{25} = 65-12 \sqrt{25-25} = 65.0\) degrees Fahrenheit

Step 5 ：To find the average temperature over the time period \(0 \leq t \leq 25\), integrate the temperature function over the time period and divide by the length of the time period, which gives \(\frac{1}{25} \int_{0}^{25} (65-12 \sqrt{25-t}) dt = 24.999999999999996\) degrees Fahrenheit

Step 6 ：Final Answer: The room's temperature when \(t=0\) is \(\boxed{5.0}\) degrees Fahrenheit, when \(t=16\) is \(\boxed{29.0}\) degrees Fahrenheit, and when \(t=25\) is \(\boxed{65.0}\) degrees Fahrenheit. The room's average temperature for \(0 \leq t \leq 25\) is approximately \(\boxed{25.0}\) degrees Fahrenheit