Working with calculus Assignment Essay Example for Free

Working with calculus Assignment Essay The nightmare has come to pass. All of Kelleys extensive surgeries and nasal passage scrapings have (unfortunately) gone awry, and he waits in the Ear, Nose, and Throat doctors office waiting area spewing bloody snot into a conical paper cup at the rate of 4 in3/min. The cup is being held with the vertex down (all the better to pool the snot in, my dear). The booger catcher has a height of 5 inches and a base of 3 inches. How fast is the mucous level rising in the cup when the snot is three inches deep? Investigating the problemÃ'Ž 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. Lets hope the doctor is on time today! Here are the formulae used to generate the tableHere 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 betterÃ'Ž