Coroners estimate time of death using the rule of thumb that a body cools about 2 degrees F during the first hour after death and about 1 degree F for each additional hour. Assuming an air temperature of 70 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)=70+286e−kt

For what value of k will the body cool by 2 degrees F in the first hour?

To find the value of k that will cause the body to cool by 2 degrees Fahrenheit in the first hour, we need to substitute the given information into the equation T(t) = 70 + 286e^(-kt) and solve for k.

First, let's consider the information given: the body cools by 2 degrees Fahrenheit in the first hour. According to the rule of thumb mentioned, we can conclude that at t = 1 hour, T(1) will be equal to 70 + (98.6 - 2). Therefore, T(1) = 166.6 degrees Fahrenheit.

Substituting these values into the equation, we get:

166.6 = 70 + 286e^(-k * 1)

We can now solve for k. Let's start by isolating e^(-k). Subtracting 70 from both sides:

96.6 = 286e^(-k)

Now, divide both sides by 286:

96.6/286 = e^(-k)

Simplifying further:

0.337 = e^(-k)

To isolate k, we take the natural logarithm of both sides:

ln(0.337) = ln(e^(-k))

Using the logarithmic property ln(e^x) = x, we have:

ln(0.337) = -k

Finally, to solve for k, multiply both sides by -1:

k = -ln(0.337)

Using a calculator or a math software, we can evaluate this expression:

k ≈ 1.085

Therefore, the value of k that will cause the body to cool by 2 degrees Fahrenheit in the first hour is approximately 1.085.