A can of soda is placed inside a cooler. As the soda cools, its temperature Tx  in degrees Celsius is given by the following function, where x  is the number of minutes since the can was placed in the cooler. =Tx+−624e−0.045x Find the initial temperature of the soda and its temperature after 20
A can of soda is placed inside a cooler. As the soda cools, its temperatureÂÂ
Tx
 in degrees Celsius is given by the following function, whereÂÂ
x
 is the number of minutes since the can was placed in the cooler.
=Tx+−624e−0.045x
Find the initial temperature of the soda and its temperature afterÂÂ
20
 minutes.
Round your answers to the nearest degree as necessary.
The Correct Answer and Explanation is :
The temperature of the soda is modeled by the equation:
[
T(x) = 24 + 6e^{-0.045x}
]
where:
- ( T(x) ) is the temperature of the soda in degrees Celsius at time ( x ) minutes after it was placed in the cooler,
- ( e ) is the base of the natural logarithm,
- ( x ) is the number of minutes since the soda was placed in the cooler.
1. Finding the initial temperature of the soda (( T(0) )):
To find the initial temperature, substitute ( x = 0 ) into the equation.
[
T(0) = 24 + 6e^{-0.045(0)}
]
[
T(0) = 24 + 6e^0
]
[
T(0) = 24 + 6(1)
]
[
T(0) = 24 + 6 = 30^\circ C
]
So, the initial temperature of the soda is 30°C.
2. Finding the temperature after 20 minutes (( T(20) )):
Now, substitute ( x = 20 ) into the equation to find the temperature after 20 minutes.
[
T(20) = 24 + 6e^{-0.045(20)}
]
[
T(20) = 24 + 6e^{-0.9}
]
We need to calculate ( e^{-0.9} ). Using a calculator:
[
e^{-0.9} \approx 0.4066
]
Now, substitute this value back into the equation:
[
T(20) = 24 + 6(0.4066)
]
[
T(20) = 24 + 2.4396
]
[
T(20) \approx 26.44^\circ C
]
Rounding to the nearest degree:
[
T(20) \approx 26^\circ C
]
Thus, the temperature after 20 minutes is 26°C.
Summary:
- The initial temperature of the soda is 30°C.
- The temperature of the soda after 20 minutes is approximately 26°C.
This equation follows the cooling process of the soda where the temperature decreases over time, and the exponential term ( e^{-0.045x} ) causes the rate of cooling to slow down as the soda approaches the temperature of the cooler.
Now, I’ll generate an image of the graph that shows how the temperature of the soda changes over time.
Here is the graph showing the temperature of the soda over time. As you can see, the temperature starts at 30°C and decreases as time progresses. After 20 minutes, the temperature of the soda has dropped to approximately 26°C. The exponential decay curve represents the cooling process, where the rate of temperature change slows down over time.
