A thermometer reading 9 degree C is brought into a room with

A thermometer reading 9 degree C is brought into a room with a constant temperature of 37 degree C. If the thermometer reads 17 degree C after 4 minutes, what will it read after being in the room for is minutes? For 11 minutes? After 5 minutes, the thermometer will read degree C. (Do not round uned the final answer. Then round to the nearest hundredth as needed.)

Solution

This requires Newton\'s Law of Cooling. The formula is:
u(x) = T + (I - T)*e^(kx)
where u(x) represents the temperature of the object at time x, I is the initial temperature of the object, and T is the temperature of the surrounding medium. k is a cooling rate determined by the physical nature of the object, the container holding the object, etc.

Plugging in our information:

u(x) = 37 + (9 - 37)*e^(kt)
==> u(x) = 37 - 28*e^(kt)

Since the temperature is 17 C at 4 minutes, then we can solve for k as follows:

17 = 37 - 28e^(4k)
==> e^(4k) = 20/28
==> 4k = ln(20/28)
==> k = ln(20/28)/4 -0.0841

Then, the equation for the temperature at any time t in minutes is:
u(x) = 37 - 28*e^(-0.0841*t)
At 6 minutes:
u(x) = 37 - 28*e^[-0.0841(4)] 17°C

At 11 minutes:
u(x) = 37 - 28*e^[-0.0841(11)] 25.9°C

I hope this helps!