Math modeling - room tempature change

Question

Let y represent the temperature (in °F) of an object in a room whose temperature is kept at a constant 60°. The object cools from 100° to 90° in 10 minutes.

A) Find an equation to model this system using y to represent temperature and t to represent minutes.

B) How much longer will it take the object to decrease to 80° ?


C) Find the temperature of the object after 15 minutes.

Answer 1

Here's an example from a Calculus class, using Newton's Law of Cooling

Example
A hard-boiled egg at 98^o C is put into a sink of 18^o C water. After 5 minutes, the egg’s temperature is 38^o C. Assuming that the water in the sink increased negligibly in temperature, how much longer does it take for the egg’s temperature to become 20^o C?
Formula: where

Solution: We find how long it takes the egg to cool from 98 to 20, and then subtract the five minutes that already elapsed.
T – 18 = (98 – 18)e^-kt
T = 18 + 80e^-kt ← Equation A.

To find k, we use T = 38 when t = 5:
which leads to

The egg’s temp 20^o is substituted into Equation A:
20 = 18 + 80e-^0.28t which leads to t = 13 min.

After subtracting the earlier elapsed 5 minutes, we have the answer: 8 min.