You pour a cup of coffee. when its poured, it is 130 F above room temperature, so we will say that difference D=130 F. 3 minutes later, it has cooled to D=117 F above room temperature. As the coffee cools, the instantaneous rate of change of D with respect to time t minutes is directly proportional to D. Write a differential equation which states this relationship.

To write the differential equation that represents the relationship between the instantaneous rate of change of D (difference in temperature) with respect to time t, we will use the fact that the rate of change is directly proportional to D.

Let's define the rate of change of D with respect to t as dD/dt. The given information states that the rate of change is directly proportional to D, so we can write this as:

dD/dt = k * D

Where k is the constant of proportionality. To solve for the differential equation, we need to determine the value of k.

We can use the information given regarding the initial and final temperatures to find k. Initially, when t = 0, D = 130°F. After 3 minutes, D = 117°F.

Using these values, we can set up an initial value problem:

dD/dt = k * D
D(0) = 130

To solve for k, we will substitute the initial condition into the differential equation:

k * 130 = dD/dt (at t = 0)

Now, we can integrate both sides of the equation:

∫(1/D) dD = ∫k dt

This simplifies to:

ln|D| = k * t + C

Where C is the constant of integration. By solving for C using the initial condition:

ln|130| = k * 0 + C
C = ln|130|

The final differential equation that relates the rate of change of D with respect to t is:

ln|D| = k * t + ln|130|

Or we can rewrite it using exponential form:

D = Ae^(k*t)

Where A = 130 and k is the constant of proportionality that can be determined based on the specific rate of cooling of the coffee.

dT/dt = -C(T - To) ,

where C is a positive constant and To is room tempoerature.

C is approximately 3.3 degrees per minute per degree C, but you would have to solve the differential equation and fit the data point get the exact value of C.