On a hot Monday morning, while people are working inside, the air conditioner keeps the temperature inside the building at 72 ^{circ} F At 3 P.M. the air conditioner is turned off, and the people go

a. The outside temperature function can be represented by a linear function, since it decreases at a constant rate from 98{eq}^{circ} {/eq}F at 3 P.M to 75{eq}^{circ} {/eq}F at midnight. Using the point-slope form, we have: {eq}M(t) = -frac{98-75}{9} (t - 3) + 98 {/eq} Simplifying, we get: {eq}M(t) = -frac{23}{9} t + 122 {/eq} b. The temperature inside the building can be modeled by a first-order exponential decay function, since it cools down after the air conditioner is turned off. Using the formula for exponential decay, we have: {eq}T(t) = T_f + (T_i - T_f) e^{-frac{t}{tau}} {/eq} where {eq}T_i = 72 {/eq}F is the initial temperature, {eq}T_f = 75 {/eq}F is the final temperature, and {eq}tau = 3 {/eq} hours is the time constant. Substituting these values, we get: {eq}T(t) = 75 + (72 - 75) e^{-frac{t}{3}} {/eq} Simplifying, we get: {eq}T(t) = 75 - 3 e^{-frac{t}{3}} {/eq} c. The temperature inside the building will reach its maximum when the rate of cooling is equal to zero. Differentiating the temperature function with respect to time, we get: {eq}frac{dT}{dt} = e^{-frac{t}{3}} {/eq} Setting this to zero and solving for t, we get: {eq}t = 0 {/eq} which means that the maximum temperature is reached immediately after the air conditioner is turned off, at {eq}t = 0 {/eq}. Substituting this value into the temperature function, we get: {eq}T(0) = 75 - 3 e^{-frac{0}{3}} = 72 {/eq}F Therefore, the maximum temperature inside the building is 72{eq}^{circ} {/eq}F, and it is reached immediately after the air conditioner is turned off.