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JP
Numerade Educator

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Problem 38 Hard Difficulty

In Exercise 9.2.28 we discussed a differential equation that models the temperature of a $ 95^oC $ cup of coffee in a $ 20^oC $ room. Solve the differential equation to find an expression for the temperature of the coffee at time $ t. $

Answer

$T=20+75 \cdot e^{-0.02 t}$

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Video Transcript

solution off problem 38 from exercise 28 in section 9.2. We know that the differential equation the capital t by these multi is equal minus four point or two multiplied by capital T minus 20 here capital t his temperature at a time. Small tea. The capital t by this multi is equal minus four point or two multiplied by capital T minus 20. So decommitted lt Divided boy T minus 20 is equal minus opened or two ds multi. Integrate both sites to get Lynn T Chemical. T minus 20 is equal minus or going toe toe. Small t plus Len Si frieze is a power. Does it these e on both sides. We ought to get e to the power. Lynn Capital T minus 20 is equal to hey to the power minus four point or two. Small tea plus linsi So e to the power. Lynn Chemical T minus 20 is equal toe e to the power Lindsay multiplied boy e to the power minus a point or two. T Using is a property. He it was a bower. Lynn A is equal. A so de minus 20 is equal toe see multiplied by E to the power miners, or point both multi So capital T is equal to 20 plus c multiplied boy e to the power miners open toes multi From the initial condition. Initial temperature is equal to 95 Phylicia's from the question. Therefore, Carrie Tell T is equal to 95 when small T is equal to zero. So, Mr to the initial conditions to get 95 is equal to 20 plus C multiplied by E to the power zero. So 95 minus 20 is equal to see multiplied by one so C is equal to 9. 75. So institutes a value off see to get difficulty is equal toe 20 plus 75 multiplied by E to the power minus 4.2 multi. So there is a will be capital T is equal to 20 plus 75 multiplied by e those of our negative over in tow small