Consider a cooling cup of coffee whose initial temperature is 205°. The room temperature is held at 70°. Suppose k = 1/16. Let y be the temperature, and y' its time derivative.

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I have the differential equation:
y' = (-1/16)(y-70)
Initial condition: yo = 205
Continuous solution equation:
y(t) = 70 + 135*exp((-1/16)(t))

I am supposed to use Euler's Method to find the temperature after 5, 20, 30 and 60 minutes. I have to find: Temperature
Euler's Method (dt = 5) and Temperature
Euler's Method (dt = 1) for 5, 20, 30 and 60. How do I do this??? Please help

http://www.math.montana.edu/frankw/ccp/calculus/des/euler-formulas/learn.htm

for dt=5, n= 135/5

for dt=1, n= 135

Euller was a dictator...

To use Euler's Method to approximate the temperature at different time points, you can follow these steps:

1. Start with the initial condition: y(0) = 205.

2. Set a time step size, dt, which represents the time interval between each approximation. In this case, you need to find the temperature at 5, 20, 30, and 60 minutes, so dt can be set to either 5 or 1.

- For dt = 5, the number of steps, n, would be 135/5 = 27.
- For dt = 1, the number of steps, n, would be 135.

3. Iterate using Euler's Method to approximate the temperature at each time point:

a) Set up a loop that runs for n steps.

b) At each step, calculate the slope of the differential equation using the equation y'(t) = (-1/16)(y-70), with the current temperature value, y, and the room temperature, 70.

c) Update the temperature using the formula: y(t + dt) = y(t) + dt * y'(t).

4. Keep track of the temperature values obtained at each time point.

Let's see how this works for both dt = 5 and dt = 1.

For dt = 5:

- At t = 5 minutes:
- Iterate 5 / dt = 1 step (n = 1).
- Calculate the slope: y'(t) = (-1/16)(y - 70) = (-1/16)(205 - 70) = -8.125.
- Update the temperature: y(5 + 5) = y(5) + 5 * (-8.125) = 205 - 40.625 = 164.375.

- At t = 20 minutes:
- Iterate 20 / dt = 4 steps (n = 4).
- Calculate the slope: y'(t) = (-1/16)(y - 70) = (-1/16)(164.375 - 70) = -6.265625.
- Update the temperature: y(20 + 5) = y(20) + 5 * (-6.265625) = 164.375 - 31.328125 = 133.046875. (Repeat this process 3 more times).

- At t = 30 minutes:
- Iterate 30 / dt = 6 steps (n = 6). Calculate and update the temperature similarly.

- At t = 60 minutes:
- Iterate 60 / dt = 12 steps (n = 12). Calculate and update the temperature similarly.

For dt = 1, follow the same steps as above but with n = 135.

This way, you can obtain the approximate temperature at each desired time point using Euler's Method.