A can of soda is placed inside a cooler. As the soda cools, its temperature (+) in degrees Celsius is given by the following function, where x is the number of minutes since the can was placed in the cooler: T(x) = -19 + 41e^(0.051x) Find the initial temperature of the soda and its temperature after 18 minutes. Round your answers to the nearest degree as necessary. Initial temperature: Temperature after 18 minutes:
The Correct Answer and Explanation is:
Here are the correct answers and the explanation.
Initial temperature: 22 °C
Temperature after 18 minutes: -2 °C
Explanation
The problem provides a function, T(x) = -19 + 41e^(-0.05x), which describes the temperature of a can of soda in degrees Celsius after a certain number of minutes, x, inside a cooler. We need to find two specific values: the initial temperature and the temperature after 18 minutes.
1. Finding the Initial Temperature
The “initial temperature” refers to the temperature at the very beginning, when the can is first placed in the cooler. In the context of the function, this corresponds to a time of zero minutes. Therefore, to find the initial temperature, we need to calculate T(0) by substituting x = 0 into the equation.
The calculation is as follows:
T(0) = -19 + 41e^(-0.05 * 0)
T(0) = -19 + 41e^(0)
A fundamental rule in mathematics is that any non-zero number raised to the power of 0 is equal to 1. So, e^0 = 1.
T(0) = -19 + 41(1)
T(0) = -19 + 41
T(0) = 22
The initial temperature of the soda is 22°C. Since this is a whole number, no rounding is necessary.
2. Finding the Temperature after 18 Minutes
To find the temperature after 18 minutes, we substitute x = 18 into the function.
The calculation is:
T(18) = -19 + 41e^(-0.05 * 18)
First, we calculate the value of the exponent:
-0.05 * 18 = -0.9
Now, we substitute this back into the equation:
T(18) = -19 + 41e^(-0.9)
Using a calculator, we find the value of e^(-0.9):
e^(-0.9) ≈ 0.40657
Next, we multiply this value by 41:
T(18) ≈ -19 + 41(0.40657)
T(18) ≈ -19 + 16.66937
Finally, we perform the addition:
T(18) ≈ -2.33063
The problem asks to round the answer to the nearest degree. The value -2.33063 is closer to -2 than to -3. Therefore, the temperature after 18 minutes, rounded to the nearest degree, is -2°C.
