Question

Coroners estimate time of death using the rule of thumb that a body coois about 2 degrees F during the first hour affer death and about 1 degree F for each additional hour. Assuming an air temperature of 58 degrees F and a living body temperature of 98.6 degrees F, the temperature T(t) in degrees F of a body at a time t hours since death is given by T(t)=58+40.6e-ᵏᵗ 1. For what value of k will the body cool by 2 degrees F in the first hour?

Answer 1

To find the value of k where the body cools by 2 degrees F in the first hour, use the equation T(t) = 58 + 40.6e^-kt with the conditions given, leading to k = -ln(0.95).

Step-by-step explanation:

To find the value of k for which the body will cool by 2 degrees F in the first hour after death, we can use the given temperature model T(t) = 58 + 40.6e-kt. We need to solve for k when the temperature has dropped by 2 degrees from the initial body temperature of 98.6°F after 1 hour (t = 1).

The initial temperature is 98.6°F, so the temperature after 1 hour should be 98.6°F - 2°F = 96.6°F. Using this information, we can set up our equation:

96.6 = 58 + 40.6e-k(1)

Subtract 58 from both sides of the equation:

38.6 = 40.6e-k

Divide both sides by 40.6:

0.95 = e-k

Now, we take the natural logarithm of both sides to solve for k:

ln(0.95) = -k

Solve for k:

k = -ln(0.95)

Calculating the right side gives us the value of k we are looking for.