Construct an exponential function from given information and then use the function to solve a contextual problem.

Question

The evening cleaning crew at a large firm discovers a dead body in the early morning hours. Police are immediately dispatched to the scene. Upon arrival at 5:00 am, police measure the temperature of the body to be 88.2°F. An hour later, the coroner arrives and again measures the temperature of the body. This time, the temperature is 86.7°F. The environmental system of the building maintains a constant temperature of 67°F.

Complete the following, including detailed explanations of your work.

1.      Use the exponential topics to determine a formula that models the temperature B of the body as a function of the time t in hours since the police arrived. Round values to 2 decimals as needed.

2.      Use your function to estimate the time of death, to the nearest half-hour.

Answer 1

If the decrease in body temp is exponential then it will be in the form of y= ab^x where y is body temp at any given time x after 5Am , a is body temp at 5Am when police arrive (starting value) and b is decay ratio

We have 2 data points t=88.2 at time x=0 and t=86.7 at t=1 hour later

we have the equation 88.2= ab^0 since b^0 =1 a=88.2

and 86.7=ab^1

86.7= 88.2b. Solve for b=86.7/88.2= .983

At time of death body temp would be 98.6 so

98.6=88.2(.983)^x. Now solve for x

(.983)^x= 98.6/88.2

(.983)^x =1.118 now take the log of both sides I will use natural log‘ can use logbase10 also

ln(.983)^x =ln1.118 By property of logs

xln(.983) = ln1.118

x= ln1.118/ln.983

x=-6.5 hours so person died 6 1/2 hours prior to 5AM which would be 10:30 PM