Question

Directions: Use Euler's method to approximate the solution curve of the differential equation in each problem. Choose a value of h and justify that this value gives accurate results using the guideline given in the text. 5.2.1 Consider a cylindrical can with radius r filled with water which drains out through a small hole in the bottom of the can. Let h(t) denote the height of the water in the can (in inches) at time t (in seconds). Using a principle called Torricelli's law, it can be shown Euler's Method 157 that h(t) is described by the differential equation πr² dt/dh =−k √ h where k>0 is a constant. If r=1,k=0.372, and the initial height is 8 inches, graph the approximate solution curve. At approximately what time is the can empty?

Answer 1

Using Euler's method with an appropriate step size, we can approximate the emptying time of a cylindrical can by graphing the height over time until it reaches zero.

Step-by-step explanation:

The differential equation given by the question is πr² dt/dh = -k √ h. Setting r=1 and k=0.372, and given an initial height h(0)=8 inches, we can apply Euler's method to approximate the solution.

To start Euler's method, we rearrange the equation to find dt/dh and then choose a suitable step size 'h'. A smaller value of 'h' will give a more accurate result, so we justify our choice of 'h' based on the guideline that the result should be close to the exact solution within a reasonable amount of computation steps.

Using Euler's method with the chosen step size, we can iterate from the initial condition, decrementing the height at each step and calculating the corresponding time until the can is empty (when h reaches 0).

By graphing the approximated values, we can find at which time the can is approximately empty by observing when the height reaches zero on our plot.