Expert: Paul Klarreich - 5/17/2008

Question
Hello I am having trouble with the following question and I was wondering if you can help me.
The temperature of an ingot of silver is 60 degrees Celsius above room temperature right now.  Twenty minutes ago, it was 70 degrees Celsius above room temperature. How far above room temperature will the silver be
(a) 15 minutes from now?
(b) 2 hours from now?
(c) When will the silver be 10 degrees Celsius above room temperature?

I know you have to use the equation P = Ce^kt.  I am confused about what you have to plug in.
I appreciate you taking the time to help me. Thank You!

Answer
Questioner:   Jessica
Category:  Calculus
Private:  No
 
Subject:  Exponential Growth and Decay
Question:  Hello I am having trouble with the following question and I was wondering if you can help me.
The temperature of an ingot of silver is 60 degrees Celsius above room temperature right now.  Twenty minutes ago, it was 70 degrees Celsius above room temperature. How far above room temperature will the silver be
(a) 15 minutes from now?
(b) 2 hours from now?
(c) When will the silver be 10 degrees Celsius above room temperature?

I know you have to use the equation P = Ce^kt.  I am confused about what you have to plug in.
I appreciate you taking the time to help me. Thank You!
.........................................
Hi, Jessica,

You have to interpret the statements and questions in terms of
P = current temperature
C = temp at time zero.
k = decay constant.

Now, then, to find your C and your k, you will need two FACTS:

The temperature of an ingot of silver is 60 degrees Celsius above room temperature right now.  
 
P = 60 at  t = 0

Twenty minutes ago, it was 70 degrees Celsius above room temperature.

P = 70 at  t = -20.

Now you can write two equations:

60 = C exp(k (0))   [A]
70 = C exp(k (-20)) [B]

A gives you  

60 = C exp(0) = C (1) = C,  so C = 60.

Now  B is:

70 = 60 exp(-20k)

7 = 6 exp(-20k)

-20k = ln(7/6)

k = - ln(7/6) / 20

So your equation is:

P = 60 exp(- ln(7/6) t / 20)

Now you can do your questions.  I'll leave most of it to you.

a) How far above room temperature will the silver be 15 minutes from now?

--- What is  P at t = 15?

b) How far above room temperature will the silver be 2 hours from now?

--- What is P at t = 120?

(c) When will the silver be 10 degrees Celsius above room temperature?

--- Find t when P = 10.