It is a dark clear night. The air temperature is 15◦ C. A body is discovered at

midnight. Its temperature is 23◦ C. One hour later, the body has cooled to 20◦ C. Use Newton’s
law of cooling to determine the time of death.

The temperature T is the ambient temperature plus a steadily decreasing difference. In this case,

T(t) = 15 + (23-15)e^(-kt)
You can see that
T(0) = 15 + (23-15)*1 = 23
that is the starting temperature. Now we are told that T(1) = 20:

15+8e^-k = 20
k = 0.47

So, T(t) = 15+8e^(-0.47t)

When did death occur? When the body had normal temperature: 37°C

15+8e^(-0.47t) = 37
t = -2.1523

So, 2.1523 hours before midnight, death arrived. That is, at

9:50:52 pm

To determine the time of death using Newton's Law of Cooling, we need to find the cooling constant (k) and then use it to calculate the time elapsed since the body's temperature was equal to the ambient temperature.

Newton's Law of Cooling is expressed as:

T(t) = T_m + (T_i - T_m) * exp(-kt)

where:
T(t) is the temperature of the body at time t,
T_m is the ambient temperature (15◦ C in this case),
T_i is the initial temperature of the body (23◦ C at midnight),
k is the cooling constant,
t is the time elapsed since the body's initial temperature was recorded.

From the given information, we can create two equations:

1) At midnight (t = 0):
23 = 15 + (23 - 15) * exp(-k * 0)

2) One hour later (t = 1):
20 = 15 + (23 - 15) * exp(-k * 1)

To solve these equations, we can subtract equation 1 from equation 2:

20 - 23 = (23 - 15) * (exp(-k) - 1)

-3 = 8 * (exp(-k) - 1)

Next, divide both sides by 8 and simplify:

-3/8 = exp(-k) - 1

Now, add 1 to both sides:

5/8 = exp(-k)

To isolate exp(-k), take the natural logarithm on both sides:

ln(5/8) = -k

Finally, solve for k:

k ≈ -0.470

Now that we have the value of k, we can use it to determine the time of death by plugging it back into one of the original equations and solving for t. Let's use equation 2:

20 = 15 + (23 - 15) * exp(-(-0.470) * t)

5 = 8 * exp(0.470t)

Dividing both sides by 8:

5/8 = exp(0.470t)

Taking the natural logarithm on both sides:

ln(5/8) = 0.470t

Finally, solve for t:

t ≈ 0.411 hours or approximately 24.7 minutes

Therefore, the estimated time of death is around 24.7 minutes before the body was discovered at midnight.