A metal bar whose temperature is 20 ˚C is placed in boiling water. The rate of change of the temperature in the bar is proportional to the difference in temperature between the bar and the surroundings. How long will it take for the bar to heat to 99.9 ˚C if the temperature of the bar after one minute of heating is 51.5 ˚C?

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This is a problem about temperature changes proportional to the difference in temperatures, based on Newton's Law of Cooling. We solve it by finding the proportionality constant 'k', and then using it to calculate the time needed for the bar to reach 99.9˚C.

Explanation:

This problem is about temperature changes, which in this case are proportional to the difference in temperature between the metal bar and the surrounding boiling water. It's based on Newton's law of cooling, but it also applies to heating. Here's how we solve it:

First, we equate the rate of change of temperature to a proportionality constant 'k' times the difference in temperatures. We then integrate from the initial temperature (20˚C) to some temperature after t minutes (51.5˚C after 1 minute). Solving this equation gives us the value of 'k'.

Next, we use this 'k' value in a similar equation, where we now integrate from initial temperature (20˚C) to the final temperature (99.9˚C). Solving this will give us the time it takes for the bar to reach 99.9˚C.

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JohnCothran JohnCothran · Answer 2 · 2023-07-25T10:45:09-04:00

Final answer:

The problem can be solved using Newton's Law of Cooling, where the rate change in temperature of an object is proportional to the temperature difference with its surroundings. From this law, a proportionality constant could be calculated and used to determine the time it would take to heat the bar to 99.9°C.

Explanation:

The subject question relates to a concept known as Newton's Law of Cooling (or heating). It states that the rate of change of the temperature is proportional to the difference in temperature between the object and the surrounding environment.

At time t=0, the temperature of the metal bar is 20°C, let's symbolize it as T0. After t=1 minute, the temperature becomes 51.5°C (T1). The temperature of the surrounding boiling water is 100°C (Ts).

We know from Newton's Law of Cooling, that T(t) = Ts + (T0 - Ts) * e^(-kt), where k is a proportionality constant. We can solve for k using the values at t=1 minute. Thus, 51.5 = 100 + (20-100)*e^(-k*1). This can be simplfied to solve for k.

Now, with the calculated value of k, we can set T(t) to 99.9 and solve for t, to get the time it takes for the temperature of the bar to reach 99.9°C.

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