2. A semi-circular trough has length L and radius R. It is filled with water to a height h from the top, as depicted in the figure below. 2R L h The volume of water V and the height h are related by the expression, V = L L[R² arcsin - h√√R² - h² (2) For a given V, the nonlinearity of this expression makes it impossible to solve for h exactly. Numerical techniques must be employed in such scenarios. For a trough with dimensions L = 10 and R = 1, write a MATLAB livescript to compute h to six decimal places, given that the trough is filled to 90% of its capacity. You are to use (i) the bisection method, (ii) the Regula-Falsi method. To begin, you must rearrange equation (2) into the form f(h) = 0. You must then split the interval [0, R] into subintervals of length 0.1, and use the Intermediate Value Theorem to find in which subin- terval the zero exists. The bracketing methods above are then to be employed on that subinterval. The example output below is for 70% capacity and three decimal place accuracy. Use it to guide the format of your output. Both root finding methods will begin on the interval [0.200000, 0.300000] The bisection method steps = 1, epsilon = steps - = 2, epsilon 0.2350, c = 0.2500 0.2507, c = 0.2250 steps 3, epsilon = 0.0074, c = 0.2375 steps = = 4, epsilon 0.1139, c = 0.2437 steps 5, epsilon = 0.0533, c = 0.2406 6, epsilon = 0.0229, c = 0.2391 = steps = steps = steps = 7, epsilon = 0.0077, c = 0.2383 8, epsilon = 0.0002, c = 0.2379 With a tolerance of le-03, after 8 steps the root is h = 0.238 The regula-falsi method = steps 1, epsilon = 0.0060, c = 0.2382 steps = 2, epsilon = 0.0000, c = 0.2379 With a tolerance of le-03, after 2 steps the root is h = 0.238
2. A semi-circular trough has length L and radius R. It is filled with water to a height h from the top, as depicted in the figure below. 2R L h The volume of water V and the height h are related by the expression, V = L L[R² arcsin - h√√R² - h² (2) For a given V, the nonlinearity of this expression makes it impossible to solve for h exactly. Numerical techniques must be employed in such scenarios. For a trough with dimensions L = 10 and R = 1, write a MATLAB livescript to compute h to six decimal places, given that the trough is filled to 90% of its capacity. You are to use (i) the bisection method, (ii) the Regula-Falsi method. To begin, you must rearrange equation (2) into the form f(h) = 0. You must then split the interval [0, R] into subintervals of length 0.1, and use the Intermediate Value Theorem to find in which subin- terval the zero exists. The bracketing methods above are then to be employed on that subinterval. The example output below is for 70% capacity and three decimal place accuracy. Use it to guide the format of your output. Both root finding methods will begin on the interval [0.200000, 0.300000] The bisection method steps = 1, epsilon = steps - = 2, epsilon 0.2350, c = 0.2500 0.2507, c = 0.2250 steps 3, epsilon = 0.0074, c = 0.2375 steps = = 4, epsilon 0.1139, c = 0.2437 steps 5, epsilon = 0.0533, c = 0.2406 6, epsilon = 0.0229, c = 0.2391 = steps = steps = steps = 7, epsilon = 0.0077, c = 0.2383 8, epsilon = 0.0002, c = 0.2379 With a tolerance of le-03, after 8 steps the root is h = 0.238 The regula-falsi method = steps 1, epsilon = 0.0060, c = 0.2382 steps = 2, epsilon = 0.0000, c = 0.2379 With a tolerance of le-03, after 2 steps the root is h = 0.238
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
--- Please answer the question fully that is attached ---
--- Thanks for your help in advance ---
![2. A semi-circular trough has length L and radius R. It is filled with water to a height h from the top,
as depicted in the figure below.
2R
L
h
The volume of water V and the height h are related by the expression,
V = L
L[R²
arcsin
- h√√R² - h²
(2)
For a given V, the nonlinearity of this expression makes it impossible to solve for h exactly. Numerical
techniques must be employed in such scenarios.
For a trough with dimensions L = 10 and R = 1, write a MATLAB livescript to compute h to six
decimal places, given that the trough is filled to 90% of its capacity. You are to use
(i) the bisection method,
(ii) the Regula-Falsi method.
To begin, you must rearrange equation (2) into the form f(h) = 0. You must then split the interval
[0, R] into subintervals of length 0.1, and use the Intermediate Value Theorem to find in which subin-
terval the zero exists. The bracketing methods above are then to be employed on that subinterval.
The example output below is for 70% capacity and three decimal place accuracy. Use it to guide the
format of your output.
Both root finding methods will begin on the interval [0.200000, 0.300000]
The bisection method
steps
=
1, epsilon
=
steps
-
=
2, epsilon
0.2350, c = 0.2500
0.2507, c = 0.2250
steps 3, epsilon = 0.0074, c = 0.2375
steps
=
=
4, epsilon 0.1139, c = 0.2437
steps 5, epsilon = 0.0533, c = 0.2406
6, epsilon = 0.0229, c = 0.2391
=
steps =
steps =
steps
=
7, epsilon = 0.0077, c = 0.2383
8, epsilon = 0.0002, c = 0.2379
With a tolerance of le-03, after 8 steps the root is h = 0.238
The regula-falsi method
=
steps 1, epsilon
=
0.0060, c = 0.2382
steps = 2, epsilon = 0.0000, c = 0.2379
With a tolerance of le-03, after 2 steps the root is h = 0.238](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2476f823-c3c8-40b1-886d-66e8e64f59f9%2F7ba8b999-95c9-4478-9f94-b3f685f8b1b8%2F9mahnq8_processed.png&w=3840&q=75)
Transcribed Image Text:2. A semi-circular trough has length L and radius R. It is filled with water to a height h from the top,
as depicted in the figure below.
2R
L
h
The volume of water V and the height h are related by the expression,
V = L
L[R²
arcsin
- h√√R² - h²
(2)
For a given V, the nonlinearity of this expression makes it impossible to solve for h exactly. Numerical
techniques must be employed in such scenarios.
For a trough with dimensions L = 10 and R = 1, write a MATLAB livescript to compute h to six
decimal places, given that the trough is filled to 90% of its capacity. You are to use
(i) the bisection method,
(ii) the Regula-Falsi method.
To begin, you must rearrange equation (2) into the form f(h) = 0. You must then split the interval
[0, R] into subintervals of length 0.1, and use the Intermediate Value Theorem to find in which subin-
terval the zero exists. The bracketing methods above are then to be employed on that subinterval.
The example output below is for 70% capacity and three decimal place accuracy. Use it to guide the
format of your output.
Both root finding methods will begin on the interval [0.200000, 0.300000]
The bisection method
steps
=
1, epsilon
=
steps
-
=
2, epsilon
0.2350, c = 0.2500
0.2507, c = 0.2250
steps 3, epsilon = 0.0074, c = 0.2375
steps
=
=
4, epsilon 0.1139, c = 0.2437
steps 5, epsilon = 0.0533, c = 0.2406
6, epsilon = 0.0229, c = 0.2391
=
steps =
steps =
steps
=
7, epsilon = 0.0077, c = 0.2383
8, epsilon = 0.0002, c = 0.2379
With a tolerance of le-03, after 8 steps the root is h = 0.238
The regula-falsi method
=
steps 1, epsilon
=
0.0060, c = 0.2382
steps = 2, epsilon = 0.0000, c = 0.2379
With a tolerance of le-03, after 2 steps the root is h = 0.238
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