Bahrain Polytechnic

Implementation of PBL in the Bachelor of Engineering Technology A sample assignment for the Engineering Mathematics Course.

This work has been inspired by the innovative work of Mr. Kealan Allen and Mr. John Donald in implementing Problem-Based Learning in the Mechanical Major of the BEngTech Programme at Bahrain Polytechnic. Their philosophy in decomposing Learning Outcomes from traditional thematic areas of Mechanical Engineering and re-integrating them within PBL Projects which are run in parallel or consecutively in order to achieve a realistic engineering end result has been adopted and transferred to the more theoretical area of Engineering Mathematics. Additionally, we adopt their teaching philosophy of creating Learning Outcomes that reflect both theoretical knowledge as well as valuable practical and generic or problem-solving skills.

Christakis Papageorgiou 4/1/2015

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ENB5913

Engineering Mathematics 2

Archimedes' Sphere and Cylinder Problem - a 2270 years old Problem

Dr. Christakis Papageorgiou

Introduction We will attempt to solve a problem, which was originally solved by Archimedes of Syracuse

approximately 2270 years ago. The objective is to calculate the ratio of the volume of a sphere

circumscribed by a cylinder as shown in Figure 1 below.

Figure 1: Sphere within a cylinder

PBL Learning Experience In the process of solving this problem, we hope to address a number of Learning Outcomes that must be

achieved within the Engineering Mathematics 2 course ENB5913.

Learning Outcomes:

1. Demonstrate detailed theoretical knowledge related to the following subjects in Mathematics:

a. Analytical geometry.

b. Calculus.

c. Integration techniques.

d. Trigonometric Identities.

e. Series expansions of functions (Taylor, Binomial).

f. Calculation of Limits.

g. Numerical Integration.

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2. Demonstrate basic practical skills related to the following:

a. Expressing mathematical notions using simple graphical sketches.

b. Using engineering software for programming numerical algorithms.

c. Using engineering software for visualization of mathematical concepts.

d. Using experimentation to verify the solution to a theoretical problem.

3. Demonstrate Generic and problem-solving skills as described by:

a. Verification of a solution which was obtained using alternative means.

b. Parameterization of discrete problems within an infinite continuum of problems.

c. Critical analysis of starting assumptions when attempting to solve a given problem.

d. Ability to transfer an analysis process to solve a similar problem in a different context.

We would like to achieve the following Learning Trajectory in the 3D PBL space spanned by the axes of

Theoretical knowledge (1), Practical Skills (2) and Generic/Problem Solving Skills (3).

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Parameterization of the solution and analytic geometry It is common practice in solving mathematical problems, to propose a solution that is elegant and

achieves a maximum end result. For our problem, this translates to the following requirement:

Obtain the volume of both the sphere and the cylinder with a single calculation.

In order to achieve this requirement we need to parameterize our problem with respect to a varying

parameter. Ideally, if we vary this parameter within its pre-defined range of variation we should be able

to obtain both solutions (sphere and cylinder) for our problem.

Task 1: Verify that the parameterization given below will include both solutions. For what

value of n we get the sphere and for what value of n we get the cylinder?

Figure 2: Defining coordinate axes

x-axis

y-axis

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Calculus

Task 2: Calculation of volume of solid objects

After parameterizing the solution using the parameter n, we can proceed with the volume calculation of

our object. We will use integration to achieve this. We will make the assumption that we can

decompose the volume of our object into an infinite number of cylinders with height dy and with

varying radius. We will then add the infinite number of elementary cylinder volumes to obtain the

volume of our object. Of course, this addition will be performed as an integration.

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Integration Techniques

Task 3: Integration by substitution

At this point, we are required to calculate the integral of Task 2. You will notice that it can be

decomposed in 3 parts, the first 2 are straightforward to calculate, the last part will require some

creativity in applying the substitution presented below:

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Trigonometric identities

Task 4: Simplification by applying trigonometric identities

Using the solution from Task 3, perform the integration and obtain the volume of the object as shown

below. Apply trigonometric identities to simplify the expression, so that it is easier to analyze it.

Task 5: Verification of the derived formula (Volume of sphere)

It is good practice to apply verification during the steps of an analysis or design procedure in order to

ensure that the proposed solution is sensible. Given the solution for the volume of the object given in

Task 4, substitute in the formula the value of n corresponding to the sphere. Thus, verify that the

formula provided in Task 3 gives the correct answer for the case when the object is a sphere of radius R.

Hopefully, you will verify that the formula given in Task 3, predicts accurately the volume of a sphere.

Taylor Series and Binomial Expansion

Task 6: Series expansions of functions

In order to calculate the volume for the case of the cylinder we will need to evaluate the expression

obtained in Task 4 as the parameter n increases to infinity. This involves taking a limit, which involves

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approximating the functions arcsin() and square root() using a series expansion and keeping only the

relevant terms of the expansion.

Limits

Task 7: Limit calculation

We need to take the limit of a function of the parameter n as the parameter n tends to infinity. We will

substitute the expressions obtained in Task 6 in the volume formula obtained in Task 4. We will do some

manipulation and then take the limit as n tends to infinity.

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Transferable mathematical skills

Task 8: Repeat the steps 1-7 above to solve a similar problem as shown below (Cone in a

Cylinder).

Critical Analysis

Task 9: Which fact from geometry have we used as a starting assumption in order to carry

out the previous analysis in solving the Problem?

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Numerical Integration as a Summation

Task 10: Discretize the formula for calculating the volume of the circumscribed object and

implement it in Matlab.

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Evaluation of the Numerical Integration Algorithm

Task 11: Use your numerical algorithm to calculate the volume of a sphere (n=1) of a radius

=10cm for increasing number of discretization points (M = 10, 100, 1000, 10000,...). Show

that the numerical algorithm will approach to the right answer as M increases to infinity.

Produce a plot similar to the one shown below.

Task 12: Can you provide a graphical sketch in order to explain why the approximate volume

is larger than the exact volume?

Task 13: Can you propose a simple experiment in order to verify the solution to Archimedes'

sphere and cylinder Problem?

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Epilogue

Archimedes solved this problem approximately 2270 years ago, without having at his disposal the

theoretical tools of calculus neither the computing power available to us today.

Extract from Wikipedia:

`...Generally considered the greatest mathematician of antiquity and one of the greatest of all

time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and

the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including

the area of a circle, the surface area and volume of a sphere, and the area under a parabola. Other

mathematical achievements include deriving an accurate approximation of pi, defining and investigating

the spiral bearing his name, and creating a system using exponentiation for expressing very large

numbers. He was also one of the first to apply mathematics to physical phenomena,

founding hydrostatics and statics, including an explanation of the principle of the lever. He is credited

with designing innovative machines, such as his screw pump, compound pulleys, and defensive war

machines to protect his native Syracuse from invasion.'

`If I have seen further than others, it is by standing upon the shoulders of giants'

Isaac Newton