Newton's Law of Cooling of pizza

But the only other point we know is the temperature of the pizza when it has already cooled down to 110 degrees. Without knowing the initial temperature, I'm not sure how we can solve for both k and t. Maybe there's some other information that we're missing?In summary, the problem presents a scenario where Bob loves vegetarian pizza and wants to know how long it will take for his pizza to cool to 110 degrees if he bakes it at 450 degrees for 20 minutes in a house with a temperature of 70 degrees. The problem provides a formula for calculating temperature at time t, but there are two unknowns (k and t) and only one known point (110 degrees). Without knowing the initial temperature of the pizza
  • #1
nineshadows
3
0

Homework Statement


Bob loves vegetarian pizza. How long will it take Bob's pizza to cool to 110 degrees if he bakes it at 450 degrees for 20 minutes? The temperature in his house is a balmy 70 degrees.


Homework Equations


T(t) = Ts + Do*e^(-k*t) where...

Do = initial temperature difference (Initial Temperature - Ts)
Ts = surrounding temperature
t = time
k = constant (cooling rate)

The Attempt at a Solution



We know the following...

> Ts := 70;

The initial temperature difference (Do) is the pizza temperature (when it came out of the oven) minus the surrounding temperature...

> Ti := 450;

> Do := Ti-Ts; ** 450 - 70 = 380 degrees **

We also know the temperature at time "t"... 110 degrees.

> T := 110;

Therefore our formula is now:

110 = 70 + 380 exp(-k*t)

** The problem is that I need to know what the cooling constant "k" is, in order to solve for the real question, "t". **

The 20 minutes given in the problem must be useful, but I'm not sure how. Can you please help?
 
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  • #2
I guess we could object to the question since we don't know the temp of the pizza before it enters the oven. Do you think that matters here?
 
  • #3
I honestly don't think this was meant to be a "trick" question... but you're right. We don't know the temperature of the pizza *before* we put it in the oven. Would this matter? I assumed (maybe wrongly) that after 20 minutes in the oven @ 450 degrees, that it would come out at that temperature. That's why I used 450 as my initial temperature (after heating in the oven).
Any other ideas?
 
  • #4
If the question does not state the initial temp before the oven, I would imagine you that you are to assume it comes out at 450 degrees.
 
  • #5
Still don't know what the heck "k" is... I have one equation and two unknowns... The formula simplifies to:

0.105 = e^(-k*t)

and I need to solve for t...
 
  • #6
I agree, had we one other point we could solve for both k and t obviously.
 

1. What is Newton's Law of Cooling?

Newton's Law of Cooling is a mathematical equation that describes the rate of change of temperature of an object as it loses heat to its surroundings. It states that the rate of cooling of an object is directly proportional to the temperature difference between the object and its surroundings.

2. How does Newton's Law of Cooling apply to pizza?

In the case of pizza, Newton's Law of Cooling can be used to predict how quickly the pizza will cool down as it sits at room temperature. As the pizza loses heat to its surroundings, its temperature will decrease until it reaches equilibrium with the room temperature.

3. Why does pizza cool down faster in the beginning?

According to Newton's Law of Cooling, the rate of cooling is proportional to the temperature difference between the object and its surroundings. In the case of pizza, when it is first taken out of the oven, its high temperature creates a larger temperature difference with the cooler room temperature, resulting in a faster rate of cooling.

4. How does the thickness of the pizza crust affect its cooling rate?

The thickness of the pizza crust can affect its cooling rate because it determines the amount of surface area available for heat to escape. A thinner crust will have a larger surface area, allowing for faster heat loss and a quicker cooling rate compared to a thicker crust.

5. Can Newton's Law of Cooling be used to reheat cold pizza?

No, Newton's Law of Cooling is only applicable for objects that are losing heat. In the case of reheating pizza, the pizza is gaining heat from its surroundings, so a different equation, such as Newton's Law of Heating, would need to be used.

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