A thermometer that has been sitting in a cold storeroom is brought into a warm kitchen. Newton's Law of Cooling can be used to model how the temperature rises in the thermometer as it approaches the kitchen's temperature: dT dt k (T - T∞) where k is the thermometer's proportionality constant and T∞ is the ambient temperature.
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The temperature initially reads 5°C when brought into the 25°C kitchen, and it reads 20°C after five minutes. Using the correct solution for temperature, when will the thermometer show approximately the correct temperature (say, 24.9°C)?
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- A bowl of soup at 80° C sits in a 20° C room and cools to 74° C after 1 minute. S Using the model for Newton's Law of Cooling T(t) = (To-Ts)ekt +Ts, where To is the initial temperature of the object and T', is the temperature of the surroundings, write the equation for the temperature of the bowl of soup after t minutes in the room. You will need to determine the value of k using the given information.A room is initially at the outdoor temperature of 25°C. Now a large fan that consumes 200 W of electricity when running is turned on. The heat transfer rate between the room and the outdoor air is given as Q˙ = UA(Ti 2 To) where U = 6 W/m2·°C is the overall heat transfer coefficient, A = 30 m2 is the exposed surface area of the room, and Ti and To are the indoor and outdoor air temperatures, respectively. Determine the indoor air temperature when steady operating conditions are established.We know from Newton's Law of Cooling that the rate at which an object warms up is proportional to the difference between the ambient temperature of the room and the temperature of the object. The differential equation corresponding to this situation is given by y' = k(M – y) where k is a positive constant. The solution to this equation is given by y = M + (yo – M)e-kt , where yo is the initial temperature of the object. Suppose your Thanksgiving turkey is kept at a temperature of 40 degrees Fahrenheit until it is put into a 350 degree Fahrenheit oven. It takes 2 hours for the turkey to warm up to a safe eating temperature of 165 degrees Fahrenheit. Find the values of yo, M, and k, for this situation, rounding your answers to 3 decimal places. Yo= M = k= Regardless of your answers above, suppose the k value in this situation is k = 0.3. Find the initial rate of increase of the turkey's temperature as soon as it is placed in the oven. Round to 3 decimal places. degrees per hour.
- A person taking a reading of the temperature in a freezer in Celsius makes two mistakes: first omitting the negative sign and then thinking the temperature is Fahrenheit. That is, the person reads – x °C as x °F . Oddly enough, the result is the correct Fahrenheit temperature. What is the original Celsius reading? Round your answer to three significant figures.Newton's Law of Cooling The temperature, u, of a heated object at a given time t can be modeled by the equation u(t) = T+ (uo - T)ekt where k<0 is a constant, T is constant ambient temperature, and wo is the initial temperature of the heated object. Suppose that an object has been heated to 58 degrees celsius and is then allowed to cool in a room that is kept at a constant 31 degrees celsius. If the temperature of the object is 46 degrees celsius after 8 minutes, when will its temperature be 37 degrees celsius? minutes.In general, the cooling of a hot steel bar can be described by the following differential equation: dT = a(T- 20) dt %3D where the surrounding room temperature is 20°C and a is a constant that describes how easily heat can transfer between the bar and the environment. Your steel bar has been heated to 200°C and will be allowed to cool from 200°C in such a way that the ODE describes its temperature. The coefficient 'a' describes how quickly heat is dissipated, and is affected by things like insulation or airflow around the bar. You wish to determine the value of 'a' required to allow the bar to cool down to below 30°C within 4 minutes, to the nearest two decimal places. Use a for loop to test all values of the 'a' coefficient between 0 and -1. Use an appropriately high resolution to ensure that your answer converges with the true answer. Answer:
- Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference C in temperature between the body and the environment around it. The temperature f(t) of the body at time t in hours after being introduced into an environment having constant temperature To is f(t) = To +Ce-kt, where C and k are constants. A cup of coffee with temperature 135°F is placed in a freezer with temperature 0°F. After 10 minutes, the temperature of the coffee is 60°F. Use Newton's Law of Cooling to find the coffee's temperature after 20 minutes. |°F. After 20 minutes the coffee will have a temperature of (Round to the nearest integer as needed.)Question 7: T (t) shows the temperature (°C) of an object at any time of t. Temperature the rate of change a = k (T - 2T), where the number k is a real number between -3 and -0.5. dt If the temperature of the object is 1000°C at the beginning and 140°C after 2 seconds, the temperature of the object is over time find the change.A solar panel receives energy from the Sun at a rate of 5.0kW. Thermal energy is transferred from the solar panel to water with an efficiency of 20%. Cold water of mass 15kg enters the solar panel every hour. The specific heat capacity of water is 4200J/(kg°C). Calculate the temperature increase of the water. temperature increase = °C
- Newton's Law of Cooling The temperature, u, of a heated object at a given time t can be modeled by the equation u(t) = T+ (uo - T)et where k < 0 is a constant, T is constant ambient temperature, and up is the initial temperature of the heated object. Suppose that an object has been heated to 90 degrees celsius and is then allowed to cool in a room that is kept at a constant 45 degrees celsius. If the temperature of the object is 73 degrees celsius after 8 minutes, when will its temperature be 58 degrees celsius? minutes.You are working as a research assistant for a professor whose research area is thermodynamics. He points out to you that Daniel Fahrenheit used the best estimate of normal human body temperature as one of the points indefining the original Fahrenheit temperature scale. On the revised scale we now use, normal human body temperature is 98.6°F. Your professor proposes a new scale on which normal human body temperature would be exactly 100°N, where the unit °N is a degree on the New scale. The temperature of freezing water would be 0°N, as on the Celsius scale. Your professor asks you to determine the following temperatures on his new scale: (a) absolute zero, (b) the melting point of mercury (-37.9°F), (c) the boiling point of water, and, for publicity at his expected future press conference, (d) the highest recorded air temperature on the Earth’s surface, 134.1°F on July 10, 1913, in Death Valley, California.A patient has an illness that typically lasts about 24 hours. The temperature, T, in degrees Fahrenheit, of the patient t hours after the illness begins is given by: T(t) = -0.012t² +0.3t +98.7. Use your calculator to graph the function and answer the following questions. Round all answers to 1 decimal place. When does the patient's temperature reach it maximum value? Answer: After hours What is the patient's maximum temperature during the illness? Answer: