Question

Can someone please help me with this problem for my Calculus class? I am not sure how to solve it and am very confused.

Answer 1

Answer:

Mr. Green is the murderer.

Step-by-step explanation:

Forget all the word problem bs. Just focus on the numbers.

You're given the equation

      $T=T_m + (T_0-T_m)e^{kt}$

$T_0=$ initial temp

$T_m=$ temp of room

$T =$ temp after t hours

$k=$ how fast the temp is changing

t = time (hours)

1.) We're given:

It took 1 hour for the body to go from 82.7° to 80.8° when the police found the body. The room temperature was 65°

$T_0=$ 82.7°

$T_m=$ 65°

$T =$ 80.8°

t = 1

$k=$ ???

the only thing we don't know is the value for k, so lets solve for it.

REFER TO STEP (1) in the picture.

You should get that  $k=ln(\frac{15.8}{17.7})$ . This is how fast the body was loosing heat.

2.) Now we know what k is (how fast the temperature is dropping), we can use it to find how long it took for his fresh body (98.6°) to drop to 82.7° when the police found it.

$T_0=$ 98.6°

$T_m=$ 65°

$T =$ 82.7°

t = ???

$k=ln(\frac{15.8}{17.7})$

We don't know what t is so lets solve for it. t will tell us how long before the police arrive (9:00pm) did the murder happened.

REFER TO STEP (2) in the picture.

You should get t=5.5079 hours. This means the murder happens 5.5 hours before 9:00pm.

3.) Since you know that the murder happened 5.5 hours before 9:00pm, subtract it.

9:00 - 5.5hours = 3:30pm

The murder happened around 3:30pm.

Mr. Green is lying because he said that he was fishing around 3:45pm-5:00pm, but that's when the murder occured.