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CREATE THE DIFFERENCE Mathematical Software with Applications This module introduces MAPLE, some powerful mathematical software to help with for solving problems. The assessments consist of solving a problem with a ‘real-world’ application. The following is a typical example.

CREATE THE DIFFERENCE Root Finding Simulation software for marine applications requires the solution to the following problem: A spherical ball of radius r is submerged to a depth d in water as shown. Assuming that the ball has a radius of 10cm and a density of 0.3, how much of the ball will be submerged?

CREATE THE DIFFERENCE Solution Archimedes’ Principle states that That is, the mass of water displaced by the ball is equal to the mass of the ball. The mass of water displaced is: The mass of the ball is: Equating gives: This simplifies to a cubic equation for d:

CREATE THE DIFFERENCE Solution Substituting r=10 andgives: Plotting using MAPLE… This is the root needed

CREATE THE DIFFERENCE Solution A root-finding algorithm, Newton-Raphson iteration, is implemented within MAPLE. Choose a starting value and use an iterative formula to ‘home in’ on the root: > x[0]:=10; > for n from 0 to 5 do > x[n+1]:=x[n]- subs(d=x[n],f(d))/subs(d=x[n],diff(f(d),d)); > od;

CREATE THE DIFFERENCE Solution This generates a sequence of approximations to the root, each better than the last: Hence the ball floats submerged to a depth of cm

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