(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

When a cake is removed from an oven, its temperature is measured at 300 degrees F, 3 min later its 200, how long will it take for it to cool off to 75? room temp is 70. Assume newtons law of cooling applies. The rate of cooling is proportional to the difference between the current temp and the ambient temp.

2. Relevant equations

This is what i did:

dx/dt = k(x-70)

3. The attempt at a solution

dx/dt = k(x-70)

integral 1/(x-70) dx = integral k dt

ln|x-70| = kt+c

after some algebra....

x = C*e^(kt) +70 where C = + or - e^c

at x(o) = 300 so...

300 = C*e^(k*0) +70

which gives me C = 230

x(t) = 230*e^(kt) +70

x(3) = 200

200 =230*e^(k*3) +70

k = 1/3 * ln(13/23)

now that i have an equation i can solve for the time it takes to get to 75

i'll leave k as k since its confusing when put into the equation

75 = 230*e^(kt) +70

i solved and got

t = (3*ln(5/230))/(ln(13/23))

which is approximately 20 minutes

Did i do this right? or did i go wrong somewhere, i just want to make sure im getting the right idea with these kinds of problems

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# Homework Help: Differential Equations: word problem

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