**Exponential Law of Heating / Cooling** **Oct 09, 7:22:16 PM** --- After sitting on a shelf for a while, a can of soda at room temperature (72°F) is placed inside a refrigerator and slowly cools. The temperature of the refrigerator is 35°F. Newton’s Law of Cooling explains that the temperature of the can of soda will decrease proportionally to the difference between the temperature of the can of soda and the temperature of the refrigerator, as given by the formula below: \[ T = T_a + (T_0 - T_a) e^{-kt} \] - \( T_a \) = the temperature surrounding the object - \( T_0 \) = the initial temperature of the object - \( t \) = the time in minutes - \( T \) = the temperature of the object after \( t \) minutes - \( k \) = decay constant The can of soda reaches the temperature of 56°F after 20 minutes. Using this information, find the value of \( k \), to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the can of soda, to the nearest degree, after 115 minutes. Here's the transcription and explanation as it might appear on an educational website: --- **Understanding the Cooling Equation** The temperature of an object cooling in a surrounding environment can be modeled using the following equation: \[ T = T_a + (T_0 - T_a)e^{-kt} \] Where: - \( T_a \) = the temperature surrounding the object - \( T_0 \) = the initial temperature of the object - \( t \) = the time in minutes - \( T \) = the temperature of the object after \( t \) minutes - \( k \) = decay constant **Problem Scenario** A can of soda reaches a temperature of 56°F after 20 minutes. Given this information, your task is to find the value of \( k \), the decay constant, accurate to the nearest thousandth. Once you have determined \( k \), use the equation to calculate the temperature of the soda after 115 minutes, rounded to the nearest degree. **Instructions** 1. Use the information provided to solve for \( k \). 2. Substitute \( k \) back into the equation. 3. Calculate the final temperature after 115 minutes. **Answer Box** Enter only the final temperature into the input box below. [Answer: _______] [Submit Answer Button] --- This format guides the user through understanding the temperature model, solving for the decay constant, and finally applying it to calculate the desired temperature of the soda after a specified time.