exponential decay

Question

A body was found in the basement of the Underwater Basket Weaving Building at​ 12:00 noon​ today, where the temperature is a steady 65 degrees Fahrenheit. When​ found, the core temperature was 91.8 degrees Fahrenheit. Two hours​ later, at​ 2:00 PM, the core temperature had fallen to 87.9.  Assuming that the body temperature was 98.6 at the time of​ death, use​ Newton's law of cooling to find the time of death.  

ROUND TO 2 DECIMAL

PLACES.

The time of death was about ___

hours before the body was found.

Answer 1

Since we have two times that are k hours after the time of death, we can simply create two equations and solve for r and k. For these equations, we'll use the Newton's law of cooling formula: f(t) = (98.6-65)e^-rt + 65.

(a₁) f(k) = 91.8 = 33.6e^-rk + 65

(b₁) f(k+2) = 87.9 = 33.6e^-r(k+2) + 65

Solving for these, we get

(a₂) rk = ln(33.6/26.8)

(b₂) rk + 2r = ln(33.6/22.9)

So r = ^{[ln(26.8) - ln(22.9)]}/₂ = 0.0786325

Plugging this into equation (a₂), we get k = ^{≈ln(33.6/26.8)}/_0.0786325 ≈ 2.88.

Thus, the time of death was about 2.88 hours prior (or about 9:07:27.44 AM).