Question 1138860: 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 Theo(13342)   (Show Source):
You can put this solution on YOUR website! the room temperature is 18 degrees centigrade.
the temperature changes according to the function T(t) = 80 * (1/2)^(t/30) + 18.
the y intercept is really the T(t) intercept in this equation.
that's the value of T(t) when t = 0.
in the context of this problem, that would be the temperature of the hot liquid when it is first placed in the room.
as t gets larger, the temperature changes according to the formula.
for example:
when t = 0, the temperature of the liquid is 98 degrees.
when t = 50, the temperature of the liquid is 43.198 degrees.
when t = 100, the temperature of the liquid is 25.937 degrees.
when t = 200, the temperature of the liquid is 18.787 degrees.
when t = 300, the temperature of the liquid is 18.078 degrees.
when t = 500, the temperature of the liquid is 18.001 degrees.
the temperature of the liquid will never go below 18 degrees centigrade because that's the temperature of the room.
the graph of the equation is shown below:
in the graph, i let y = T(t) and i let x = t.
as x gets larger, (1/2) ^ (x/30) gets smaller until it approaches 0
as it approaches 0, the temperature of the liquid approaches 18 degrees centigrade which is the temperature of the room.
to summarize.
when t = 0, that's the temperature of the liquid as it is placed in the room.
as t gets larger, the temperature of the liquid approaches 18 degrees centigrade and will never get lower than that.
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