T(10) = 415 F
T(20) = 347 F
Surrounding Temperature = 75 F
I have the equation T(t) = 75 + ce^(-kt) and am unsure of how to find the values for c and k due to not knowing the initial temperature
Captain Matticus, LandPiratesInc
Favorite Answer
Well, you have 2 unknowns and 2 points. Set up a system of equations, solve for c and k, then find T(0)
415 = 75 + c * e^(-10k)
340 = c * e^(-10k)
347 = 75 + c * e^(-20k)
272 = c * e^(-20k)
340 = c * e^(-10k)
272 = c * e^(-20k)
340 / 272 = c * e^(-10k) / (c * e^(-20k))
340/272 = (c/c) * e^(-10k - (-20k))
4 * 85 / (4 * 68) = 1 * e^(20k - 10k)
85/68 = e^(10k)
ln(85/68) = 10k
(1/10) * ln(85/68) = k
340 = c * e^(-10k)
340 = c * e^(-10 * (1/10) * ln(85/68))
340 = c * e^(-ln(85/68))
340 = c * e^(ln(68/85))
340 = c * (68/85)
340 * 85 / 68 = c
85 * 85 * 4 / 68 = c
85 * 85 / 17 = c
85 * 5 = c
425 = c
Of course, I just realized, while working out the problem, that 85/68 can be further reduced. I'm not too quick on my 17's table sometimes
T(t) = 75 + c * e^(-kt)
c = 425
k = (1/10) * ln(85/68) = (1/10) * ln(5/4)
T(t) = 75 + 425 * e^((-1) * (1/10) * ln(5/4) * t)
T(t) = 75 + 425 * e^((1/10) * ln(4/5) * t)
T(t) = 75 + 425 * (4/5)^(t/10)
T(0) = 75 + 425 * (4/5)^(0/10)
T(0) = 75 + 425 * 1
T(0) = 500
Initial temperature is 500 F
L. E. Gant
"c" and "k" are constants You have two temperatures (different "t") so you can calculte them using simultaneous equations
415 =75 + ce^(-10k)
and
347 = 75 + ce^(-20k)
solve for c and k