answer:1. **Given Information**: - Diameter of the disc: (1 text{ meter}) - Weight of disc with diameter (1 text{ meter}): (100 text{ kg}) - Standard deviation of radius: (10 text{ mm} = 0.01 text{ m}) 2. **Expectation and Variance of Radius**: - Mean radius, ( mathrm{E} R = frac{1}{2} text{ m} = 0.5 text{ m} ) - Variance of radius, ( mathrm{Var}(R) = (0.01 text{ m})^2 = 10^{-4} text{ m}^2 ) 3. **Mathematical Expectation of the Area of One Disc**: - The area of a disc, ( S ), with radius ( R ) is given by ( S = pi R^2 ). - We need to find the expectation of the area, ( mathrm{E}[S] = mathrm{E}[pi R^2] ). Using the property ( mathrm{E}[R^2] = mathrm{Var}(R) + (mathrm{E} R)^2 ): [ mathrm{E}[R^2] = 10^{-4} + (0.5)^2 = 10^{-4} + 0.25 = 0.2501 ] Therefore, [ mathrm{E}[S] = pi mathrm{E}[R^2] = pi cdot 0.2501 ] 4. **Mathematical Expectation of the Mass of One Disc**: - Given the standard disc ( pi (0.5)^2 text{ m}^2 ) weighs (100 text{ kg}), each unit area weighs (frac{100 text{ kg}}{0.25pi}). Since ( mathrm{E}[S] = 0.2501 pi ): [ mathrm{E}[text{mass}] = frac{0.2501 pi cdot 100 text{ kg}}{0.25 pi} = 100.04 text{ kg} ] 5. **Expected Mass of a Stack of 100 Discs**: - If one disc weighs, on average, (100.04 text{ kg}): [ text{Expected mass of 100 discs} = 100 times 100.04 text{ kg} = 10004 text{ kg} ] 6. **Comparison with Engineer Sidorov's Prediction**: - Engineer Sidorov predicted the stack would weigh (10000 text{ kg}). - The actual expected mass is (10004 text{ kg}). - The error in Engineer Sidorov's estimate is: [ 10004 text{ kg} - 10000 text{ kg} = 4 text{ kg} ] # Conclusion: [ boxed{4 text{ kg}} ]

answer:Let's denote the variable as ( x ). According to the problem, if you multiply ( x ) by 5 and then by 7, you get 5 and 4/9, which can be written as a mixed number ( 5 frac{4}{9} ) or as an improper fraction ( frac{49}{9} ). So the equation we can set up is: [ 7 times (5 times x) = frac{49}{9} ] First, we can simplify the left side of the equation by multiplying 7 and 5 together: [ 35x = frac{49}{9} ] Now, to solve for ( x ), we divide both sides of the equation by 35: [ x = frac{49}{9 times 35} ] [ x = frac{49}{315} ] To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 7: [ x = frac{49 div 7}{315 div 7} ] [ x = frac{7}{45} ] Now, to convert this fraction to a decimal, we can perform the division: [ x = 7 div 45 ] [ x approx 0.155555... ] Rounded to two decimal places, the variable ( x ) is approximately: [ x approx boxed{0.16} ]

answer:We need to determine the location of the individuals A, B, C, and D in an art gallery such that they satisfy the given visual constraints. 1. **Placement of A:** - Given that A cannot see anyone, A must be located such that no other people are in A's line of sight. - The only position that satisfies this is the topmost position, ensuring there are no other individuals within A's view. 2. **Placement of B:** - B can only see C. - For B to see only one person, B must be positioned such that there is a clear line of sight to C but not to A or D. - The appropriate position that allows B to see only C is the bottom right. 3. **Placement of C:** - C can see both B and D. - Thus, C must be positioned to have both B and D within line of sight simultaneously. - This can only be achieved if C is placed in the bottom left, i.e., diagonally opposite B to see B and possibly across the viewing path towards D. 4. **Placement of D:** - D can only see C. - D must be positioned such that C is within D's line of sight. - The suitable position for D is directly to the left, ensuring D can see C without any other individuals within sight. Thus, when all conditions are satisfied with the above placements, the position of P (located in mathrm{C}'s position) is clearly identified. Conclusively, P should be filled with: [ boxed{C} ]

answer:We first rewrite the right-hand side: [cos^5 x - sin^5 x = frac{cos x - sin x}{sin x cos x},] thus, multiplying through by (sin x cos x), we get: [sin x cos x (cos^5 x - sin^5 x) = cos x - sin x.] This can be factored into: [sin x cos x (cos x - sin x)(cos^4 x + cos^3 x sin x + cos^2 x sin^2 x + cos x sin^3 x + sin^4 x) - (cos x - sin x) = 0.] Expanding and simplifying the factor with trigonometric identities (using (sin^2 x + cos^2 x = 1)), we get: [cos x sin x (cos x - sin x)(1 + cos x sin x - cos^2 x sin^2 x) - (cos x - sin x) = 0.] Let (q = cos x sin x), the equation becomes: [q (cos x - sin x)(1 + q - q^2) - (cos x - sin x) = 0,] which factors to: [-(cos x - sin x)(q - 1)^2 (q + 1) = 0.] Since (|q| = frac{1}{2}|sin 2x| le frac{1}{2}), (q) can't be 1 or -1, so (cos x = sin x), or (tan x = 1). Solutions in ([0^circ, 360^circ]) are (45^circ) and (225^circ), and their sum is (boxed{270^circ}).