Question

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)=Ce(−kt)+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

Answer 1

23

Explanation:

data

`f(15)= 40\\t= 15\\ C=?\\t_0=0`

`t_0=0` because it's the initial time, when we start counting

To know how much the temperature is worth in the initial time
`t_0=0` we must find out the value of the constant C with the data we have of the situation at 15 minutes

`f(t)= Ce^((-kt))+14\\t=15\\f(15)= Ce^((-k(15)))+14\\40= Ce^(-((0.045)(15)))+14\\40-14= Ce^(-0.675)\\(26)/(e^(-0.675)) = C \\8.89=C`

Find the initial temperature by replacing the data given and obtained

`f(t)= Ce^((-kt))+14\\t=0\\f(0)= Ce^((-k(0)))+14\\f(0)= 8.89(1)+14\\f(0)= 22.89\\f(0)= 23`

Answer 2

65

Explanation:

You can use f(15) = 40 to solve for C, then find f(0), the initial temperature.

40 = f(15)

40 = Ce^(-0.045·15) +14 = .50916C +14

26 = .50916C

26/.50916 = C ≈ 51.065

Then f(0) is ...

f(0) = 51.065·e^0 +14 = 65.065 ≈ 65

The initial temperature of the water was 65 degrees Fahrenheit.