At 10: 30 am, detectives discover a dead body in a room and measure its temperature at 31∘c. one hour later, the body’s temperature had dropped to 24.8∘c. determine how many hours had passed since the death occured (when the body temperature was a normal 37∘c), by the time the body was discovered, assuming that the temperature in the room was held constant at 20∘c. (use decimal notation. give your answer to one decimal place. round any intermediate calculations, if needed, to no less than three decimal places.)

The reference time is 10:30 AM,

The body temperature is 31 ° C,

The room temperature () is 20 ° c,

Newton and the law of cooling are useful in this situation, the equation says,

Where k is proportionality constant

t= time

T= Temperature of Body,

Thus,

We can integrate here,

We have our first equation,

When the measurement t = 0 was made, the temperature T = 31 ° c

11=e^c

We have know that

Our equation now is,

When t=1hour then T=24.8, then

Then our equation is now,

Finally, we know can find the time when T=37, so

-0.83t=ln\frac({17}{11})

Therefore the time is 10:30-31min, we have that at 9:59AM the body was still alive