# Thread: Exponential Growth and Logarithms

1. ## Exponential Growth and Logarithms

So, I have these two problems for Calculus that I'm rather stuck on right now.

The information given for the first is this:

(Exponential growth)
Colony weight:
15 mg at 9:00 A.M.
25 mg at 11:30 A.M.
45 mg at ?

where 9:00 A.M. corresponds to t=0.

What's given for the next one is this:

"In the cold storage room, which is maintained at a constant temperature of 20° C, Detective Marino discovered the body of Dr. Expo’s assistant, Nat Log. The Detective measured the temperature of the body to be 34° C at 5:00 p.m. Two hours later, when Dr. Kay Scarpetta, the chief medical examiner, arrived to examine the bodies and evidence, she measured Nat’s body temperature to be 31.7° C.
Dr. Scarpetta knew that Newton’s Law of Cooling says that the temperature of a cooling object is given by the function
T
(t)=Sekt+M

where M is the temperature of the surrounding environment, although she didn’t know either S or k. She did know that the normal body temperature of a living person is 37° C. "

And I'm supposed to find the missing times for both. Any help would be greatly appreciated!

2. Originally Posted by Rumor
The information given for the first is this:

(Exponential growth)
Colony weight:
15 mg at 9:00 A.M.
25 mg at 11:30 A.M.
45 mg at ?

where 9:00 A.M. corresponds to t=0.
you should already be acquainted with the general equation for uninhibited exponential growth ...

$\displaystyle y = y_0 e^{kt}$

$\displaystyle y_0 = 15$ mg

$\displaystyle y = 15e^{kt}$

at $\displaystyle t = 2.5$ hrs, $\displaystyle y = 25$ mg

$\displaystyle 25 = 15e^{2.5k}$

solve for $\displaystyle k$ ... then use the completed equation to find the time when $\displaystyle y = 45$ mg

3. 1.

A(t) = A(0) e^(kt)

A(0) =15

A(t) = 15 e^(kt)

A(2.5) = 25 =15 e^(2.5k)

5/3 = e^(2.5k)

ln(5/3) = 2.5 k

solve for k and you'll Have A(t)

Set A(t) = 45 and solve for t with ln

2. T(t) = se^(-kt) + 20

T(0) = 34 use this to solve for s

T(2) =31.7 use this to solve for k

You now have T(t) set this equal to 37 to determinine time of death

t will be negative the time before the body was first found

4. Many thanks to the both of you!