A yam is put in a 180∘C oven and heats up according to the differential equation?
A yam is put in a 180∘C oven and heats up according to the differential equation
dH/dt = -k(H-180), for k a positive constant
If the yam is at 10∘C when it is put in the oven and after 40 minutes the temperature of the yam is 90∘C, solve the differential equation.
H= ? Celsius
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Verified answer
dH/dt = -k(H-180)
Separate the variables.
dH/(H-180) = -kdt
Integrate.
ln|H-180| = c - kt where c is a constant.
Substitute the initial conditions.
ln|10-180| = c - 0
c = ln(170)
ln|H-180| = ln(170) - kt
Substitute the other known value.
ln|90-180| = ln(170) - 40k
40k = ln(170) - ln(90)
40k = ln(17/9)
k = (1/40)ln(17/9)
Solution:
ln|H-180| = ln(170) - (t/40)ln(17/9)
180-H = 170(9/17)^(t/40) . . . because H<180
H = 180 - 170(9/17)^(t/40)