Question
Jeremy boiled one cup of water and then let it cool at room temperature. The table shows the temperature, in degrees Fahrenheit, of the water after a given number of five-minute intervals. Jeremy used an exponential function to model the temperature w(t), in degrees Fahrenheit, of the water after t intervals.
Interval
0
1
2
3
4
Temperature
210
167
145
131
121
Interval
0
1
2
3
4
Temperature
210
167
145
131
121
Answers
GPT 3.5
To find the exponential function that models the temperature of the water after t intervals, we can use the form of an exponential function:
w(t) = ae^(bt)
where w(t) is the temperature after t intervals, a is the initial temperature (when t = 0), e is the base of the natural logarithm (approximately 2.71828), and b is a constant.
From the table, we can see that the initial temperature (when t = 0) is 210 degrees Fahrenheit. Therefore, a = 210.
To find the constant b, we can use the fact that the temperature decreases by a constant factor each time interval:
210e^(b) = 167
167e^(b) = 145
145e^(b) = 131
131e^(b) = 121
Solving these equations simultaneously gives us b ≈ -0.126.
Therefore, the exponential function that models the temperature of the water after t intervals is:
w(t) = 210e^(-0.126t)
w(t) = ae^(bt)
where w(t) is the temperature after t intervals, a is the initial temperature (when t = 0), e is the base of the natural logarithm (approximately 2.71828), and b is a constant.
From the table, we can see that the initial temperature (when t = 0) is 210 degrees Fahrenheit. Therefore, a = 210.
To find the constant b, we can use the fact that the temperature decreases by a constant factor each time interval:
210e^(b) = 167
167e^(b) = 145
145e^(b) = 131
131e^(b) = 121
Solving these equations simultaneously gives us b ≈ -0.126.
Therefore, the exponential function that models the temperature of the water after t intervals is:
w(t) = 210e^(-0.126t)