Question

The Law and Order detectives enter a house at 11:30 am and found a man on the floor murdered. They take his body temperature finding to be 91.8 F. The body is taken to the morgue and at 12:30 pm the body temperature is 84.4 F. Using an exponential model for the body temperature, to the nearest minute, what time was he killed. Normal body temperature is 98.6.

Answer 1

Time of murder = 10:39 am

Explanation:

Let the equation of exponential function representing the final temperature of the body after time 't' is,

f(t) =
`a(e)^(nt)`

Here, a = Initial temperature

n = Constant for the change in temperature

t = Duration

At 11:30 am temperature of the body was 91.8°F.

91.8 =
`98.6(e)^(nt)` --------(1)

Time to reach the body to the morgue = 12:30 pm

Duration to reach = 12:30 p.m. - 11:30 a.m.

= 1 hour

Therefore, equation will be,

84.4 =
`91.8(e)^(n* 1)`

eⁿ =
`(84.4)/(91.8)`

ln(eⁿ) = ln(0.9194)

n = -0.08403

From equation (1),

91.8 =
`98.6(e)^(-0.08403t)`

`(e)^(0.08403t)=(98.6)/(91.8)`

`ln[(e)^(0.08403t)]=ln[(98.6)/(91.8)]`

0.08403t = 0.07146

t = 0.85 hours

t ≈ 51 minutes

Therefore, murder was done 51 minutes before the detectives arrival.

Time of murder = 11:30 - 00:51

= 10:90 - 00:51

= 10:39 am