SOLUTION: For this problem in Newtons law of cooling, what does the e mean? Why do we use ln and what is ln?
T (t) = Te + (T0 − Te ) e^-kt
160= 69 + (190-69)e^k2
160-69 = 69-69
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-> SOLUTION: For this problem in Newtons law of cooling, what does the e mean? Why do we use ln and what is ln?
T (t) = Te + (T0 − Te ) e^-kt
160= 69 + (190-69)e^k2
160-69 = 69-69
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Question 658621: For this problem in Newtons law of cooling, what does the e mean? Why do we use ln and what is ln?
T (t) = Te + (T0 − Te ) e^-kt
160= 69 + (190-69)e^k2
160-69 = 69-69(121)e^k2
91 = (121)ek2
91/121 = (121/121)e^2k
e^2k = 91/121
2k = ln(91/121)
k = .5(ln(91/121))
k = .5(-.284931)
k = -.1425
69 + (180 - 69)e^-.1425t = 130
69-69 + 121e-.1425t = 130-69
121e.-1425t = 61
e^-.1425t = 61/121
-.1425t = ln(61/121)
t = (-.684917)/(-.1425)
t = 4.8 minutes Found 2 solutions by stanbon, MathLover1:Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! For this problem in Newtons law of cooling, what does the e mean? Why do we use ln and what is ln?
T (t) = Te + (T0 − Te ) e^-kt
T(t) is he temperature of the object at time "t".
Te is the temperature of the environment.
To is the temperature of the object at some initial time (e.g. time of death)
-----
The e in e^kt is the irrational number e = 2.718281828...
------------------------------
160= 69 + (190-69)e^k2
160-69 = 69-69(121)e^k2
91 = (121)ek2
91/121 = (121/121)e^2k
e^2k = 91/121
2k = ln(91/121)
---
ln is the "natural log"; it is the power of "e" that gives you some
particular number.
For example:
ln(e^k2) = k2
ln(5) = 1.609.. because e^(1.609..) = 5
--------------------------------
------------
k = .5(ln(91/121))
k = .5(-.284931)
k = -.1425
69 + (180 - 69)e^-.1425t = 130
69-69 + 121e-.1425t = 130-69
121e.-1425t = 61
e^-.1425t = 61/121
-.1425t = ln(61/121)
t = (-.684917)/(-.1425)
t = 4.8 minutes
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Cheers,
Stan H.
You can put this solution on YOUR website! The number is an important mathematical ,
approximately equal to , that is the of
the logarithm ln.
is important simply because it has all those nice properties
you've been studying. Whenever you take the derivative of
(that's to the ), you get back again
It's the only function on Earth that will do that (except things
like and variants like that). That's pretty cool stuff.
functions are written as ,
where a represents the rate of the exponential change.
In such cases where exponential changes are involved we usually use
another kind of logarithm called .
The can be thought of as . What this means is that it is a logarithmic operation
that when carried out on raised to gives
us the .
This logarithm is labeled with ln (for "natural log") and its definition is: .
(