A thermometer is taken from a room at $55^{\circ} \mathrm{F}$ to the outdoors, where the temperature is $10^{\circ} \mathrm{F}$. The reading on the thermometer drops to $40^{\circ} \mathrm{F}$ after one minute.
a. Find the reading on the thermometer after (i) three minutes, (ii) 8 minutes, and (iii) one hour.
b. How long will it take for the reading to drop to $21^{\circ} \mathrm{F}$ ?
a. (i) After three minutes, the thermometer will read $\square^{\circ} \mathrm{F}$.
(Do not round until the final answer. Then round to the nearest hundredth as needed.)

Step 1 ：First, we need to understand that the thermometer's temperature change follows an exponential decay model. The formula for this model is \(T(t) = T_{\infty} + (T_0 - T_{\infty})e^{-kt}\), where \(T(t)\) is the temperature at time \(t\), \(T_{\infty}\) is the final temperature, \(T_0\) is the initial temperature, and \(k\) is the decay constant.

Step 2 ：We know that the initial temperature \(T_0\) is 55 degrees Fahrenheit, the final temperature \(T_{\infty}\) is 10 degrees Fahrenheit, and the temperature after one minute \(T(1)\) is 40 degrees Fahrenheit. We can substitute these values into the formula to solve for \(k\).

Step 3 ：Substituting the given values into the formula, we get \(40 = 10 + (55 - 10)e^{-k}\). Simplifying this equation gives us \(30 = 45e^{-k}\).

Step 4 ：Solving for \(k\), we get \(k = -\ln\left(\frac{30}{45}\right)\).

Step 5 ：Now that we have the value of \(k\), we can substitute it back into the original formula to find the temperature at any given time \(t\).

Step 6 ：(i) To find the temperature after three minutes, we substitute \(t = 3\) into the formula. This gives us \(T(3) = 10 + (55 - 10)e^{-3k}\).

Step 7 ：(ii) To find the temperature after eight minutes, we substitute \(t = 8\) into the formula. This gives us \(T(8) = 10 + (55 - 10)e^{-8k}\).

Step 8 ：(iii) To find the temperature after one hour (or 60 minutes), we substitute \(t = 60\) into the formula. This gives us \(T(60) = 10 + (55 - 10)e^{-60k}\).

Step 9 ：For part b, we need to find the time \(t\) when the temperature drops to 21 degrees Fahrenheit. We can do this by setting \(T(t) = 21\) and solving for \(t\).

Step 10 ：Substituting \(T(t) = 21\) into the formula, we get \(21 = 10 + (55 - 10)e^{-kt}\). Solving this equation for \(t\) gives us \(t = -\frac{1}{k}\ln\left(\frac{21 - 10}{45}\right)\).

Step 11 ：Finally, we substitute the value of \(k\) we found earlier into this equation to find the value of \(t\).