Solution to problem 5.7.3 from the collection of Kepe O.E.

Solution to problem 5.7.3 from the collection of Kepe O.?. is as follows: given a system of equations of the form Ax = b, where A is a matrix of size n x n, x and b are vectors of length n. It is required to find a solution to this system.

To solve the problem, you can use the Gauss-Jordan method or the LU decomposition method. The first method consists in constructing an extended matrix of the system, reducing it to a stepwise form and reversing it, in which the values ​​of the unknowns are sequentially found from the upper rows of the matrix. The second method is based on decomposing the matrix A into the product of two matrices L and U, where L is a lower triangular matrix with ones on the diagonal, U is an upper triangular matrix. After this, solving the system is reduced to sequentially solving two systems of equations Ly = b and Ux = y.

The choice of method depends on the specific task and the properties of the matrix A.

***

Problem 5.7.3 from the collection of Kepe O.?. is formulated as follows:

"The height CD is lowered onto straight line AB. Find the distance from point E, which lies on segment CD, to the middle of segment AB, if AB = 10 cm and CD = 6 cm."

To solve this problem, it is necessary to use the property of right triangles, namely: the height lowered onto the hypotenuse divides it into two smaller hypotenuses, one of which is equal to the projection of the other onto the same hypotenuse.

Thus, to solve the problem, you need to find the length of the segment CE, which is the projection of the height CD onto the hypotenuse AB. This can be done by knowing that triangles AEC and BDC are similar to each other, since their corresponding angles are equal (angle AEC is equal to angle BDC, since they are vertical angles, and angle ACE is equal to angle BCD, since they are corresponding angles) . Also, from the similarity of triangles it follows that the ratio of the lengths of the sides is equal to the ratio of the lengths of the hypotenuses:

AE/BD = EC/DC

We substitute the known values ​​and get:

AE/BD = EC/6

AE/(10 - AE) = EC/6

EC = 6AE/(10 - AE)

Then we find the distance from point E to the middle of AB, which is equal to half the hypotenuse AB, that is, 5 cm.

So, to find the required distance, you need to calculate the length of the segment EC using the formula above and then calculate the distance between point E and the middle AB, which is equal to 5 cm minus the length of the segment EC.

Problem 5.7.3 from the collection of Kepe O.?. is formulated as follows:

A homogeneous horizontal beam DE weighing G = 6 kN at point D rests on a horizontal curved rod ABC, held by a vertical cable CF. The distance BD between points B and D is 1 m. It is necessary to determine the distance CD from point C to point D, at which the tension cable CF will be equal to 1 kN.

Answer to the problem: 2 meters.

Solution to problem 5.7.3 from the collection of Kepe O.?. is as follows:

Given a set of points on a plane. It is necessary to find a triangle with vertices at these points that has the smallest area.

To solve this problem, you can use an algorithm for enumerating all possible triangles formed by three points from a given set. For each triangle, its area is calculated, and the triangle with the smallest area is selected.

However, to speed up the search process, you can use algorithms for calculating the convex hull and triangulating a set of points on a plane.

The algorithm for calculating the convex hull allows you to find a polygon in which all points from a given set lie on its boundary. You can then iterate over the triangles formed by the three vertices of this polygon and select the triangle with the smallest area.

The triangulation algorithm allows you to divide many points on a plane into disjoint triangles. Then, you can go through all the triangles and choose the triangle with the smallest area.

Thus, to solve problem 5.7.3 from the collection of Kepe O.?. You can use various algorithms to find the triangle with the smallest area from a given set of points on the plane.

***

1. Solution to problem 5.7.3 from the collection of Kepe O.E. is an excellent digital product for students and learners of any level.
2. I recommend the solution to problem 5.7.3 from the collection of O.E. Kepe. for anyone looking for high-quality digital material for learning mathematics.
3. Solution to problem 5.7.3 from the collection of Kepe O.E. is a user-friendly and intuitive digital product that will help you better understand math concepts.
4. I received great benefit from solving problem 5.7.3 from the collection of O.E. Kepe. and recommend it to all my friends and colleagues.
5. Solution to problem 5.7.3 from the collection of Kepe O.E. is an indispensable digital product for those who study mathematics at a professional level.
6. This digital product is presented in a convenient format and contains all the necessary materials for successfully solving problem 5.7.3 from the collection of Kepe O.E.
7. Solution to problem 5.7.3 from the collection of Kepe O.E. is an excellent choice for those who want to improve their math skills and develop logical thinking.

Peculiarities:

The solution of problem 5.7.3 was very useful for my preparation for the exam.

With this solution, I better understood the material on thermodynamics.

Very

Solution of problem 5.7.3 from the collection of Kepe O.E. is a great digital product for math students.

I would recommend solving problem 5.7.3 from O.E. Kepe's collection. those who want to improve their knowledge and skills in mathematics.

This digital product allows not only to solve the problem, but also to better understand the material presented in the collection by Kepe O.E.

Solution of problem 5.7.3 from the collection of Kepe O.E. is a convenient and fast way to test your knowledge and skills in mathematics.

This digital item is very useful for students who are preparing for exams or math tests.

I would recommend the solution of problem 5.7.3 from the collection of Kepe O.E. those who want to better understand math and improve their grades.

Solution of problem 5.7.3 from the collection of Kepe O.E. is a great digital product for those who study mathematics on their own.

This digital product will help you better understand the material and avoid mistakes when solving problems.

Solution of problem 5.7.3 from the collection of Kepe O.E. is a great way to train your mind and develop your math problem solving skills.

I would recommend solving problem 5.7.3 from O.E. Kepe's collection. anyone who wants to improve their knowledge and skills in mathematics and achieve great success in this area.

A very high quality solution to the problem, everything is step by step disassembled and understandable.

Many thanks to the author for such a detailed and accessible analysis of the problem.

Collection of Kepe O.E. - an excellent choice for those who want to understand mathematics deeper.

A very useful digital product for students and schoolchildren.

Solving the problem helped me better understand the material and prepare for the exam.

The analysis of the problem is clear even to those who are not very strong in mathematics.

A very convenient format - you can watch and repeat the analysis of the problem several times in order to remember everything.

Thanks to the author for such a detailed and useful work.

Solving the problem helped me to believe in my mathematical abilities.

I recommend this digital product to anyone who wants to improve their knowledge in mathematics.