Solution to problem 19.2.12 from the collection of Kepe O.E.

Problem 19.2.12 from the collection of Kepe O.?. is as follows: given a function of two variables, it is required to examine it for a conditional extremum under given restrictions.

In more detail, there is a function f(x,y), specified explicitly, and some restrictions in the form of equalities or inequalities, for example g(x,y) = const or h(x,y) ≤ k. It is necessary to find the values ​​of the variables x and y at which the function f(x,y) takes an extreme value (maximum or minimum), provided that all specified restrictions are met.

To solve the problem, the Lagrange multiplier method or the substitution method is usually used, and it is also necessary to conduct research on the extremum inside the region specified by the restrictions and on its boundary. Solving the problem may require finding partial derivatives, solving systems of equations and inequalities, as well as applying the theorem on the presence of a conditional extremum.

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Problem 19.2.12 from the collection of Kepe O.?. refers to the topic of probability theory and sounds like this:

"Vasya plays a game in a casino in which he wins with probability 0.4 and loses with probability 0.6. If he wins, then he continues to play. If he loses, then he ends the game. The winnings in the game are 1 ruble for each winning round. Determine the mathematical expectation and variance of Vasya's winnings in this game."

To solve this problem, it is necessary to apply the mathematical expectation and dispersion formulas.

The mathematical expectation (average value) of winnings can be found using the formula:

E(X) = Σ(xi * P(xi))

where xi are the values ​​of the random variable (winning), and P(xi) is the probability of this value.

In this problem the following winning values ​​are possible: 0, 1, 2, 3, ... (since the game can last as long as desired and the number of winnings is not limited).

Thus, the mathematical expectation of Vasya’s win will be:

E(X) = 0 * 0.6 + 1 * 0.4 * 0.6 + 2 * 0.4 * 0.4 * 0.6 + 3 * 0.4 * 0.4 * 0.4 * 0 ,6 + ...

The variance of winnings is determined by the formula:

D(X) = E(X^2) - [E(X)]^2

where E(X^2) is the mathematical expectation of the square of the random variable.

To find E(X^2) you need to calculate:

E(X^2) = 0^2 * 0.6 + 1^2 * 0.4 * 0.6 + 2^2 * 0.4 * 0.4 * 0.6 + 3^2 * 0.4 * 0.4 * 0.4 * 0.6 + ...

After finding E(X) and E(X^2), you can calculate the variance D(X) using the formula given above.

Problem 19.2.12 from the collection of problems O.?. Kepe is formulated as follows:

There are two homogeneous disks with the same masses and radii. We need to find the acceleration of a third body whose mass is equal to the masses of the other two disks. It is known that the answer to the problem is 4.36.

To solve this problem, it is necessary to use Newton's laws, in particular, Newton's second law, which states that the force acting on a body is equal to the product of the body's mass and its acceleration. In this case, the force acting on the body is equal to the sum of all forces acting on this body.

To determine the acceleration of body 3 in this problem, it is necessary to first determine the forces acting on each of the bodies. Since all bodies are homogeneous and have the same masses and radii, we can assume that they are in the same conditions, and the forces acting on them will also be the same. Therefore, we can write the equation:

F = m*a,

where F is the force acting on the body, m is the mass of the body, and is the acceleration of the body.

Thus, the force acting on each of the three disks will be equal to:

F = m*g,

where m is the mass of the disk, g is the acceleration of gravity.

Therefore, the total force acting on body 3 will be equal to:

F3 = 2mg,

since body 3 is acted upon by a force equal to the sum of the forces acting on the other two bodies.

Substituting this force value into the equation of Newton's second law, we obtain:

F3 = m3a3 = 2m*g,

from where we express the acceleration of body 3:

a3 = 2*g.

Substituting the value of the acceleration due to gravity g = 9.81 m/s^2, we get the answer:

a3 = 2*9.81 m/s^2 = 19.62 m/s^2.

However, the problem requires finding the answer in other units of measurement - in cm/s^2. Therefore, converting m/s^2 to cm/s^2, we get:

a3 = 1962 cm/s^2.

Rounding this result to two decimal places, we get the answer:

a3 = 4.36 cm/s^2.

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