A freezer has a temperature of 14 degrees Fahrenheit. An ice-cube tray full of water is placed in the freezer. The function $f(t)=C e^{(-k t)}+14$ represents the situation, where $t$ is time in minutes, $C$ is a constant, and $k=0.045$.
After 15 minutes the water has a temperature of 40 degrees. What was the initial temperature of the water? Round your answer to the nearest degree, and do not include units.
Provide your answer below:
Answer
Rounding to the nearest degree, we find that the initial temperature of the water was \(\boxed{51}\) degrees Fahrenheit.
Step 1 ：We are given the function \(f(t)=C e^{(-k t)}+14\), where \(t\) is time in minutes, \(C\) is a constant, and \(k=0.045\).
Step 2 ：We know that after 15 minutes (\(t=15\)), the temperature is 40 degrees. We can substitute these values into the function to solve for \(C\), which represents the initial temperature of the water.
Step 3 ：Substituting the given values into the equation, we get \(40 = C e^{(-0.045 * 15)} + 14\).
Step 4 ：Solving this equation for \(C\), we find that \(C\) is approximately 51.0648573752160.
Step 5 ：Rounding to the nearest degree, we find that the initial temperature of the water was \(\boxed{51}\) degrees Fahrenheit.
