When a cold drink is taken from a refrigerator, its temperature is 5°C. After 25 minutes in a 20°C room its temperature has increased to 10°C. (Round the answers to two decimal place.)
(a) What is the temperature of the drink after 60 minutes?
(b) When will its temperature be 12°C?
a) (5 - 20)e^(-0.0162t) + 20
At 60 minutes it is (-15)e^(-0.0162*60) + 20 = 14.325
b) 12 = (-15)e^(-0.0162t) + 20
-8 = -15e^(-0.0162t)
ln(8/15) = -0.0162t
t = (1/0.0162)*ln(15/8) = 38.8 minutes
Well, I have to say, that drink really needs to work on its time management skills!
(a) After 60 minutes, the temperature of the drink is approximately 14.33°C. So it's gone from cold to cool, but still not quite warm enough for a cozy night by the fire.
(b) Now, let's solve the mystery of when the temperature will hit 12°C. It looks like it'll take approximately 38.8 minutes. So, set your alarm and keep an eye on that drink! It's going to be 12°C before you know it.
(a) To find the temperature of the drink after 60 minutes, we can use the formula:
Temperature = (Initial Temperature - Room Temperature) * e^(-0.0162t) + Room Temperature
Plugging in the values:
Temperature = (5 - 20)e^(-0.0162 * 60) + 20
Temperature = (-15)e^(-0.972) + 20
Temperature ≈ 14.32°C
So, the temperature of the drink after 60 minutes is approximately 14.32°C.
(b) To find when the temperature of the drink will be 12°C, we can set up the equation:
12 = (-15)e^(-0.0162t) + 20
Solving for t, we can rearrange the equation and solve for t using logarithms:
-8 = -15e^(-0.0162t)
Taking the natural logarithm of both sides:
ln(8/15) = -0.0162t
Dividing both sides by -0.0162:
t = (1/0.0162) * ln(15/8)
t ≈ 38.8 minutes
So, the temperature of the drink will be 12°C after approximately 38.8 minutes.
To find the temperature of the drink after 60 minutes, we can use the equation:
Temperature = (Initial temperature - Room temperature) * e^(-0.0162 * t) + Room temperature
Given the initial temperature is 5°C, the room temperature is 20°C, and the time is 60 minutes, we can substitute these values into the equation:
Temperature = (5 - 20) * e^(-0.0162 * 60) + 20
Temperature = (-15) * e^(-0.972) + 20
Temperature ≈ 14.33°C (rounded to two decimal places)
Therefore, the temperature of the drink after 60 minutes is approximately 14.33°C.
To determine when the temperature will reach 12°C, we set up the equation:
12 = (5 - 20) * e^(-0.0162 * t) + 20
Simplifying this equation, we get:
-8 = -15 * e^(-0.0162 * t)
To isolate the exponential term, we divide both sides by -15:
8/15 = e^(-0.0162 * t)
Taking the natural logarithm (ln) of both sides, we have:
ln(8/15) = -0.0162 * t
Finally, dividing both sides by -0.0162 and solving for t, we get:
t = (1/0.0162) * ln(15/8)
t ≈ 38.8 minutes (rounded to two decimal places)
Therefore, the temperature of the drink will reach 12°C approximately 38.8 minutes after being taken out of the refrigerator.