Investigating the problem
The volume of a cone V = where r is the radius of the cone and he is its height
For the full cone or any part of it, the ratio of r:h remains fixed, so
As we are only interested in the rate of change of the height we need to eliminate r so use r = 3h/10 for all levels
So the new V = so to find
h3 = and h =
So making a table to find for t= 0 to 25 and hence work out roughly how long the cone takes to fill up, and the height value at each stage and also radius each time. As can be seen, the full height and radius is reached at about t < 15 minutes. Let’s hope the doctor is on time today!
Here are the formulae used to generate the table.
Here is the graph of h and r against time:
Both h and r increase rapidly in the 1st 5 minutes before the rate of increase slows as t increases.
Using Numerical methods
Various rates of change could be investigated, including the rate of change of h with respect to V, the rate of change of r with respect to t and so on. However, the question asks about the rate of change of h with respect to t, so this will be investigated using the Leibnitz formula : to estimate gradients using a spreadsheet. The following graph was obtained:
As can be seen this graph of the rate of change of height (the speed at which height changes) is not very helpful, as there is a lot of change for t = 0 to t = 2 but after that the rate of change is much less. Some investigation shows that most of the change takes place between t = 0 and t = 1. So tracing the rate of change of the 2 sections on different graphs, with the one involving the first section in much more detail, will give a better picture.
The table:
And the graph
The reduction in speed of the heights rise is very marked
The table for t = 1 to t = 14:
and the graph:
The question requires the rate of change at h = 3. From the table this can be see between t = 2 and t = 4, where the gradient is between 0.46 and 0.29 inches per minute
Using differentiation
V = so and we were also told
So using the Chain Rule: =
Filling what is known: 4 = so
So when h = 3 = 0.393 inches per minute
Conclusion:
The numerical method does not give a very accurate result and provided the Chain rule is used, the calculus method is much better
