answer:Let's denote the temperatures on Monday, Tuesday, Wednesday, Thursday, and Friday as M, T, W, Th, and F, respectively. According to the information given: (M + T + W + Th) / 4 = 48 (T + W + Th + F) / 4 = 46 We also know that M = 43 degrees. Now, let's calculate the total temperature for Monday, Tuesday, Wednesday, and Thursday: (M + T + W + Th) = 48 * 4 (43 + T + W + Th) = 192 Now, let's express the total temperature for Tuesday, Wednesday, Thursday, and Friday: (T + W + Th + F) = 46 * 4 (T + W + Th + F) = 184 We can now set up an equation to solve for F by subtracting the first total from the second total: (T + W + Th + F) - (M + T + W + Th) = 184 - 192 F - M = -8 Substitute the value of M (43 degrees) into the equation: F - 43 = -8 F = -8 + 43 F = 35 Therefore, the temperature on Friday was boxed{35} degrees.

answer:To determine which figure is definitely a symmetrical figure, we examine each option individually: - **A: Parallelogram** A parallelogram has opposite sides that are equal and parallel but does not necessarily have symmetry along any axis. Therefore, it cannot be guaranteed to be a symmetrical figure. Rightarrow Parallelogram does not meet the criteria. - **B: Trapezoid** A trapezoid, having only one pair of parallel sides, does not guarantee symmetry. The lengths and angles can vary in such a way that symmetry is not achieved. Rightarrow Trapezoid does not meet the criteria. - **C: Circle** A circle is symmetrical about any diameter and any line through its center, making it a figure with infinite axes of symmetry. Rightarrow Circle meets the criteria and is definitely a symmetrical figure. - **D: Triangle** A triangle may or may not be symmetrical, depending on its type (equilateral, isosceles, or scalene). Only certain conditions (like being isosceles or equilateral) make a triangle symmetrical. Rightarrow Triangle does not meet the criteria. Given the analysis above, the figure that is definitely a symmetrical figure is: boxed{C} Circle.

answer:This is an arithmetic series where the first term ( a = -41 ) and the common difference ( d = 3 ). We are asked to find the sum up to the final term ( ell = 7 ). 1. **Determine the number of terms ( n )**: We start by determining ( n ) using the equation for the ( n )-th term of an arithmetic sequence: [ a + (n-1)d = ell ] Substituting ( a = -41 ), ( d = 3 ), and ( ell = 7 ), [ -41 + (n-1) cdot 3 = 7 ] Solving for ( n ), [ (n-1) cdot 3 = 48 implies n-1 = 16 implies n = 17 ] 2. **Calculate the sum ( S )**: The sum of an arithmetic series is given by: [ S = frac{n}{2} cdot (a + ell) ] Substituting ( n = 17 ), ( a = -41 ), and ( ell = 7 ), [ S = frac{17}{2} cdot (-41 + 7) = frac{17}{2} cdot (-34) = -17 cdot 17 = boxed{-289} ] Conclusion: Hence, the sum of the series (-41) + (-38) + cdots + 7 is ( boxed{-289} ).

answer:The function f(x) = arcsin(log_{m}(2nx)) is defined when: [-1 leq log_{m}(2nx) leq 1,] which translates to: [frac{1}{m} leq 2nx leq m.] Expressing in terms of x, we get: [frac{1}{2mn} leq x leq frac{m}{2n}.] The length of the interval is then: [frac{m}{2n} - frac{1}{2mn} = frac{m^2 - 1}{2mn},] which equals: [frac{m^2 - 1}{2mn} = frac{1}{2027}.] Solving for n, we have: [n = frac{2027(m^2 - 1)}{2m} = frac{2027m^2 - 2027}{2m}.] We seek to minimize n+m, which is: [n + m = frac{2027m^2 - 2027 + 2m^2}{2m} = frac{2029m^2 - 2027}{2m}.] Since m must divide 2027 and m^2 - 1 is relatively prime to m, we find the smallest m that divides 2027. Factoring 2027 gives 2027 = 7 times 17 times 17. The smallest divisor greater than 1 is 7. For m = 7: [n = frac{2027(7^2 - 1)}{7} = frac{2027 cdot 48}{7} = 13848,] thus the smallest possible value of m+n is: [7 + 13848 = boxed{13855}.]