**Heating Baby's Milk Using Newton's Law of Cooling**Bob heats up a bottle of milk for his baby, Jill, every morning. Jill will only drink the bottle if it is at 98 degrees. So Bob places the bottle in a cup of hot water (170 degrees). The bottle starts off at 35 degrees. After 2 minutes, the bottle is 80 degrees. Using Newton's law of cooling, which states that the rate of change in temperature, \( H \), is proportional to the difference between the object and the surrounding temperature, we can determine how long Bob should leave the bottle in the hot water. Assume that the water stays at 170 degrees.Concepts Covered:1. Newton's Law of Cooling2. Temperature Change and Time Calculation3. Practical Application in Daily LifeWe’ll employ Newton's law of cooling formula:\[ \frac{dT}{dt} = -k(T - T_{\text{env}}) \]Where:- \( T \) = temperature of the object at time \( t \)- \( T_{\text{env}} \) = surrounding temperature (170 degrees in this case)- \( k \) = constant of proportionality- \( t \) = timeSteps:1. Establish initial conditions. - Initial temperature of the bottle, \( T_0 \) = 35 degrees - Surrounding temperature, \( T_{\text{env}} \) = 170 degrees - Temperature after 2 minutes, \( T(t=2) \) = 80 degrees2. Use these conditions to solve for \( k \).3. Calculate the time required for the bottle to reach 98 degrees using the derived constant \( k \).By solving the differential equation with the provided initial conditions, we’ll ascertain the precise time Bob should leave the bottle in the hot water. This system of equations and practical examples makes Newton's law of cooling comprehensible and applicable to everyday scenarios.

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Bob heats up a bottle of milk for his baby, Jill, every morning. Jill will only drink the bottle if it is at 98 degrees. So Bob places the bottle in a cup of hot water (170 degrees). The bottle starts off at 35 degrees. After 2 minutes the bottle is 80 degrees. Use Newton's heating-cooling law, that the rate of change in the temperature, H, is proportional to the difference between the object and the surrounding temperature, to determine how long Bob should leave the bottle in the hot water. Assume that the water stays at 170 degrees.

Bob heats up a bottle of milk for his baby, Jill, every morning. Jill will only drink the bottle if it is at 98 degrees. So Bob places the bottle in a cup of hot water (170 degrees). The bottle starts off at 35 degrees. After 2 minutes the bottle is 80 degrees. Use Newton's heating-cooling law, that the rate of change in the temperature, H, is proportional to the difference between the object and the surrounding temperature, to determine how long Bob should leave the bottle in the hot water. Assume that the water stays at 170 degrees.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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