SOLUTION: Please help with this question with exponential functions A cup of hot liquid was left to cool in a room whose temperature was 18°C. The temperature changes according to the fun

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Question 1138844: Please help with this question with exponential functions
A cup of hot liquid was left to cool in a room whose temperature was 18°C. The temperature changes according to the function T(t)=80(1/2)^t/30+18 What are the y-intercept and the equation of the horizontal asymptote? What do they mean in this context?
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The equation you show for the temperature is not correct:

T%28t%29=80%281%2F2%29%5Et%2F30%2B18

Assuming you know something about how a hot liquid cools, you know this is not the right equation. You should know that parentheses are required:

T%28t%29=80%281%2F2%29%5E%28t%2F30%29%2B18

As with any function, the y-intercept is the value when x=0. For this equation,

T%280%29+=+80%281%2F2%29%5E%280%2F30%29%2B18+=+80%281%29%2B18+=+98

The horizontal asymptote is the value of the function when t gets very large. For this equation, since 1/2 to a very large power approaches 0,

T%28infinity%29+=+0%2B18+=+18

In this problem, it should be clear that the y-intercept 98 is the initial temperature of the liquid (in °C).

The horizontal asymptote is 18, which is the temperature in °C of the room. That makes sense; a hot liquid left in a room for a very long time will eventually have a temperature that is the same as the temperature of the room.

Note that, while the problem doesn't ask anything about it, the exponent "t/30" means that the difference between the temperature of the liquid and the room temperature gets cut in half every 30 units of time (presumably minutes).