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Chapter 6: Q. 5 (page 573)

Theorems: Fill in the blanks to complete each of the following theorem statements.

If an object with temperature T(t)is in a location with constant ambient temperature A, then the temperature of the object can be modeled by a differential equation of the form dTdt=_____, with solution T(t)=_____.

Short Answer

Expert verified

If an object with temperature T(t)is in a location with constant ambient temperature A, then the temperature of the object can be modeled by a differential equation of the form dTdt=k(A-T), with solution T(t)=A-(A-T0)e-kt.

Step by step solution

01

Step 1. Given information

If an object with temperature T(t)is in a location with constant ambient temperature A, then the temperature of the object can be modeled by a differential equation of the form dTdt=_____, with solution T(t)=_____.

02

Step 2. Filling the blanks

If an object with temperature T(t)is in a location with constant ambient temperature A, then the temperature of the object can be modeled by a differential equation of the form dTdt=k(A-T), with solution T(t)=A-(A-T0)e-kt.

The differential equation is:dTdt=k(A-T)

By separation of variables and integrating, we get,

∫dTA-T=∫kdt⇒-lnA-T=kt+C⇒lnA-T=-kt-C⇒A-T=e-kt-C⇒A-T=Be-kt⇒T=A-Be-kt

Since, T(0)=T0and T0=A-Be-k(0)

⇒B=A-T0

Therefore,T(t)=A-(A-T0)e-kt

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