"The time rate of change of the temperature of an object is proportional to the difference between the temperature surrounding it (called the ambient temperature) and the temperature of the object (which we assume is approximately uniform at all times)" 1. State a differential equation that is a mathematical model of the situation using T to represent temperature of the object, t for time, and k for your constant of proportionality. If t is temperature and Tamb is the ambient temperature then the equation is: dT k((Tamb –t) dt A. State the form of a general initial condition for the differential equation using proper notation: B. If we solve this differential equation what are we solving for? Be specific in terms of the DE you wrote. C. State one solution to the differential equation you can find by inspection:

"The time rate of change of the temperature of an object is proportional to the difference between the temperature surrounding it (called the ambient temperature) and the temperature of the object (which we assume is approximately uniform at all times)" 1. State a differential equation that is a mathematical model of the situation using T to represent temperature of the object, t for time, and k for your constant of proportionality. If t is temperature and Tamb is the ambient temperature then the equation is: dT k((Tamb –t) dt A. State the form of a general initial condition for the differential equation using proper notation: B. If we solve this differential equation what are we solving for? Be specific in terms of the DE you wrote. C. State one solution to the differential equation you can find by inspection:

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

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"The time rate of change of the temperature of an object is proportional to the difference between the
temperature surrounding it (called the ambient temperature) and the temperature of the object (which
we assume is approximately uniform at all times)"
1. State a differential equation that is a mathematical model of the situation using T to represent
temperature of the object, t for time, and k for your constant of proportionality.
If t is temperature and Tamb is the ambient temperature then the equation is:
dT
k((Tamb –t)
dt
A. State the form of a general initial condition for the differential equation using proper notation:
B. If we solve this differential equation what are we solving for? Be specific in terms of the DE
you wrote.
C. State one solution to the differential equation you can find by inspection:
D. Another solution is A+ Ce¯kt . Verify this is in fact a solution by plugging it into your
differential equation.

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According to Newton’s Law of Cooling, the rate of change of the temperature of an objectcan be modeled using the following differential equation: dT/dt=−κ(T−A) where T is the temperature of the object (in◦F), A is the room temperature (in◦F), and κ is the constant of proportionality. (a) Use an appropriate method to solve the differential equation.Remember that κ and A are constants. (b) Use your answer to (a) to solve the following: On a crime show, a dead body is discovered in a hotel room. A forensic technician recordedthat the victim’s body temperature was 91.4◦F at 6:00 pm. One hour later, the coroner arrived and found that the victim’s body temperature had fallen to 88.7◦F. If the thermostatin the hotel room is set at 68◦F, determine when the victim was murdered.(Assume the victim was a healthy 98.6◦F at the time of death.)
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Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).
Newton's Law of Cooling states that the rate at which the temperature of a cooling object decreases and the rate at which a warming object increases are proportional to the difference between the temperature of the object and the temperature of the surrounding medium. The differential equation formula is given as: dT = k (T – A) dt - T = temperature of object at time t А ambient temperature (aka temperature of the surrounding medium) k = constant of proportionality Okay, now here's the problem. A glass of delicious orange soda (with an initial temperature of 34°F) is placed in a room with a constant temperature of 72°F. One hour later, the temperature of the orange soda is 52°F. Find the exact simplified value of k. Hint: solve the differential equation for T and note that A is a constant.
For this problem T is the variable for the temperature, k is the growth constant (or constant of proportionality) and is positive.Newton's Law of cooling states that the temperature of an object changes at a rate proportional to the difference of the temperature of the surrounding medium and the temperature of the object. If the temperature of the surrounding medium is 340∘� write a differential equation to model the situation.