A can of soda is placed inside a cooler. As the soda cools, its temperature T(x) 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^(-0.45x) Find the initial temperature of the soda: Find its temperature after 20 minutes: Round your answers to the nearest degree as necessary.
The Correct Answer and Explanation is:
To solve the given problem, we need to evaluate the temperature function at two different points: at the initial time (when x = 0) and after 20 minutes (when x = 20).
The Temperature Function:
The temperature of the soda is modeled by the function:T(x)=−19+41e−0.45xT(x) = -19 + 41e^{-0.45x}T(x)=−19+41e−0.45x
Where:
- T(x)T(x)T(x) is the temperature of the soda at time xxx (in minutes),
- xxx is the number of minutes since the soda was placed in the cooler,
- eee is Euler’s number (approximately 2.71828),
- −19-19−19 represents a constant temperature shift,
- 414141 is the initial temperature difference from −19-19−19,
- −0.45-0.45−0.45 is the rate of cooling of the soda over time.
Finding the Initial Temperature (at x=0x = 0x=0):
To find the initial temperature, substitute x=0x = 0x=0 into the temperature function:T(0)=−19+41e−0.45(0)T(0) = -19 + 41e^{-0.45(0)}T(0)=−19+41e−0.45(0)T(0)=−19+41e0T(0) = -19 + 41e^{0}T(0)=−19+41e0T(0)=−19+41(1)(because e0=1)T(0) = -19 + 41(1) \quad \text{(because } e^0 = 1\text{)}T(0)=−19+41(1)(because e0=1)T(0)=−19+41=22∘CT(0) = -19 + 41 = 22^\circ CT(0)=−19+41=22∘C
So, the initial temperature of the soda is 22°C.
Finding the Temperature After 20 Minutes (at x=20x = 20x=20):
Now, to find the temperature after 20 minutes, substitute x=20x = 20x=20 into the function:T(20)=−19+41e−0.45(20)T(20) = -19 + 41e^{-0.45(20)}T(20)=−19+41e−0.45(20)T(20)=−19+41e−9T(20) = -19 + 41e^{-9}T(20)=−19+41e−9T(20)≈−19+41(0.000123)(using a calculator for e−9≈0.000123)T(20) \approx -19 + 41(0.000123) \quad \text{(using a calculator for } e^{-9} \approx 0.000123\text{)}T(20)≈−19+41(0.000123)(using a calculator for e−9≈0.000123)T(20)≈−19+0.00504T(20) \approx -19 + 0.00504T(20)≈−19+0.00504T(20)≈−18.99496T(20) \approx -18.99496T(20)≈−18.99496
Rounding to the nearest degree, the temperature after 20 minutes is approximately -19°C.
Summary:
- The initial temperature of the soda is 22°C.
- The temperature of the soda after 20 minutes is approximately -19°C.
This result illustrates the cooling process, where the temperature quickly drops initially and then slows down as the soda approaches a cooler equilibrium.
