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PROBLEM 5.19 SOLUTION

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This document presents a complete and methodical solution to Problem 5.19 from Numerical Methods for Engineers by Chapra. The work focuses on applying root‑finding numerical methods to determine the solution of a nonlinear equation specified in the problem. The document begins by restating the original mathematical model, identifying the function to be solved, and outlining the physical or engineering context in which the equation arises. The solution is developed using a structured Bisection method

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Institution
University Of South Africa (Unisa)
Course
Numerical Methods for Civil Engineers A

Content preview

DEFINE THE FUNCTION
Archimedes Principle :Weight of displaced liquid = weight of submerged part

𝜌 ∗𝑉 =𝜌 ∗𝑉

Where :

𝑉 = 𝑆𝑢𝑏𝑚𝑒𝑟𝑚𝑒𝑟𝑔𝑒𝑑 𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒

𝑉 = 𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒

𝜌 = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟

𝜌 = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒

𝜋ℎ
𝑢𝑛𝑠𝑢𝑟𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉 = (3𝑟 − ℎ)
3
𝑉 =𝑉 −𝑉
4
𝑉 = 𝜋𝑟
3
4 𝜋ℎ
𝑉 = 𝜋𝑟 − (3𝑟 − ℎ)
3 3

1000 ∗ [ 𝜋𝑟 − (3𝑟 − ℎ)] = 200 ∗ 𝜋𝑟
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Institution
University of South Africa (Unisa)
Course
Numerical Methods for Civil Engineers A
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