Question about coffee’s temperature

I want to talk about a question on yesterday’s quiz. It’s the question about the temperature of a cup of coffee. I found it’s little bit tricky and difficult to think this question correctly. The following content is my way to solve it, I really hope that some one could come up with another way, that is simpler and more normal, to think about this question.

We know that the original temperature is 180°F and the temperature inside the office is 68°F. Saying that every minutes the temperature of coffee will drop r times (coffee temperature)-(office temperature), where r = 0.022. The question is what will the temperature of the coffee be after 10 minutes.

At first I cannot figure out how to calculate the answer because the I don’t know how to represent the “coffee temperature” for it keeps changing. My friend suggested me to see the change of the temperature, which is “(coffee temperature)-(office temperature)”, as one unit and that’s how I figure it out.

Assuming the original difference of the temperature between the coffee and the office as “D”, so we know that every minutes the coffee temperature will drop 0.022 × D (°F). Because when the coffee temperature drops, the temperature difference “D” will drop the same amount of degrees, so I decided to mainly focus on the changes of the temperature difference “D” rather than the changes of the coffee temperature. Thus, after one minutes, I have:

D – 0.022 × D = (1 – 0.022)¹ × D

It appears that some rules have shown up. Keep follow it. After ten minutes, we should have:

(1 – 0.022)¹⁰ × D

now bring the value of the original temperature difference in:

(1 – 0.022)¹⁰ × (180 – 68)

= (1 – 0.022)¹⁰ × 112

≈ 90°F

That is the temperature difference after 10 minutes. However, the number we need to calculate is the temperature of coffee after 10 minutes. For the reason that in this case, the temperature of the office won’t change, so the last step of us is use the “D” after 10 minutes plus 68 degrees, which is the temperature of the office.

90 + 68 = 158°F

Basically that is how I did on this question. The fact is that I realized that I’m quite insensitive on thinking this type of question. I am little bit unfamiliar and uncomfortable with this way of thinking. If you have simpler and more normal way to solve this one, please let me know.



2 thoughts on “Question about coffee’s temperature

  1. Jayson Chang

    I thought the quiz was fairly easy, except for this one question! I tried to apply the formula we learned in class, but I couldn’t figure out a way to do so. Instead, I just do the calculation manually, which took me a solid 3 to 4 minutes to do just that one question. Your method is awesome !

  2. Karen Kuo

    It does took me awhile to get the right answer for this question. What I did was only calculate the temperature change between each minute and tried to figure out the final temperature after 10 minute. I found out that the temperature change between each minute is about 2 and 2×10=20, then 180-20=160 which is around 158 since I round the numbers for 2. Well I guess your approach is definitely more correct! Thanks for sharing!

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