Final answer:
To model the temperature of the coffee at time t, we can use an exponential decay function. By using the given initial and final temperatures, we can solve for the constant k and the initial temperature T0. The function that models the temperature of the coffee at time t is T(t) = T0 * e^(kt).
Step-by-step explanation:
To find a function that models the temperature of the coffee at time t, we need to use the concept of exponential decay. In this case, the coffee is cooling down from 200 degrees Fahrenheit to 70 degrees Fahrenheit over a period of 10 minutes.
The general form of an exponential decay function is given by: T(t) = T0 * e^(-kt), where T(t) is the temperature at time t, T0 is the initial temperature, e is the base of the natural logarithm (approximately 2.718), k is a constant, and t is the time.
Using the given information, we can plug in the values to solve for k. At t = 0 (initial time), T(t) is 200 degrees Fahrenheit. At t = 10 minutes (final time), T(t) is 150 degrees Fahrenheit.
Substituting these values into the exponential decay function, we get: 200 = T0 * e^(-k*0) (equation 1) and 150 = T0 * e^(-k*10) (equation 2).
Since e^0 = 1, equation 1 simplifies to: 200 = T0.
Dividing equation 2 by equation 1, we get: 150 / 200 = e^(-k*10).
Taking the natural logarithm of both sides, we have: ln(150/200) = -10k.
Solving for k, we can substitute the value of k back into equation 1 to find T0, the initial temperature of the coffee.
So, the function that models the temperature of the coffee at time t is: T(t) = T0 * e^(-kt).