A cup of coffee is freshly brewed to 95°C - Leaving Cert Mathematics - Question 9 - 2021
Question 9
A cup of coffee is freshly brewed to 95°C.
The temperature, T(t), in degrees centigrade, of the coffee as it cools is given by the formula
$$T(t) = Ae^{-0.08t} + 20... show full transcript
Worked Solution & Example Answer:A cup of coffee is freshly brewed to 95°C - Leaving Cert Mathematics - Question 9 - 2021
Step 1
Show that A = 75.
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Answer
To show that A = 75, we will substitute the initial temperature of the coffee into the provided formula.
At time t = 0 minutes, the initial temperature T(0) is equal to 95°C:
T(0)=Ae−0.08(0)+20T(0)=A+20
Therefore, we set up the equation:
95=A+20
Solving for A gives:
A=95−20=75
Thus, we have shown that A = 75.
Step 2
Explain what the value 20 in the formula represents in the context of the coffee cooling.
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Answer
In the context of the coffee cooling, the value 20 represents the lower limit of the coffee's temperature as it cools down. This is the temperature that the coffee will approach but never fall below, essentially representing the ambient room temperature.
Step 3
Find the decrease in the temperature of the coffee 10 minutes after brewing.
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Answer
To find the decrease in temperature after 10 minutes, we first calculate the temperature at t = 10:
Substitute t = 10 into the formula:
T(10)=75e−0.08(10)+20T(10)=75e−0.8+20
Using the approximate value of e^{-0.8} ≈ 0.449, we have:
T(10)≈75(0.449)+20T(10)≈33.675+20≈53.675
The decrease in temperature from the initial 95°C is:
Decrease=95−T(10)=95−53.675≈41.325
Rounding to the nearest whole number, the decrease in temperature is approximately 41°C.
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