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

the total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)