answer:1. **Calculate the reciprocals of the numbers**: The numbers are 2, 3, and 6.5. Their reciprocals are: [ frac{1}{2}, frac{1}{3}, text{ and } frac{1}{6.5} ] 2. **Sum the reciprocals**: [ frac{1}{2} + frac{1}{3} + frac{1}{6.5} = frac{3}{6} + frac{2}{6} + frac{1}{6.5} = frac{5}{6} + frac{1}{6.5} ] To add (frac{5}{6}) and (frac{1}{6.5}), convert (frac{1}{6.5}) to a fraction with a denominator of 6: [ frac{1}{6.5} approx frac{10}{65} = frac{2}{13} ] And find a common denominator for (frac{5}{6}) and (frac{2}{13}): [ frac{5}{6} cdot frac{13}{13} + frac{2}{13} cdot frac{6}{6} = frac{65}{78} + frac{12}{78} = frac{77}{78} ] 3. **Calculate the average of the reciprocals**: Since there are three numbers, the average of their reciprocals is: [ frac{frac{77}{78}}{3} = frac{77}{234} ] 4. **Find the reciprocal of the average**: The harmonic mean is the reciprocal of the average of the reciprocals: [ frac{1}{frac{77}{234}} = frac{234}{77} ] 5. **Conclusion**: The harmonic mean of the numbers 2, 3, and 6.5 is frac{234{77}}. The final answer is boxed{textbf{(A)} frac{234}{77}}

answer:Let's denote Trenton's fixed weekly earnings (excluding commission) as ( E ). According to the problem, Trenton earns a commission of 0.04 (or 4%) on his sales. If he must make sales of 7750 to reach his goal of earning no less than 500, then the commission from these sales would be: ( 0.04 times 7750 = 310 ) dollars. This 310 represents the commission part of his earnings for the week. Since his goal is to earn no less than 500, and we know that 310 of that comes from commission, we can find his fixed weekly earnings by subtracting the commission from his goal earnings: ( 500 - 310 = 190 ) dollars. Therefore, Trenton earns boxed{190} weekly excluding the commission.

answer:Solution: The center of the circle x^{2}+y^{2}-2x+4y=0 is C(1,-2), because the line 2x+y+m=0 passes through the center of the circle x^{2}+y^{2}-2x+4y=0, therefore the center C(1,-2) lies on the line 2x+y+m=0, therefore 2times1-2+m=0, Solving this, we get m=0. Therefore, the answer is: boxed{0}. By finding the center of the circle x^{2}+y^{2}-2x+4y=0 as C(1,-2), and then substituting the center C(1,-2) into the line 2x+y+m=0, the result can be obtained. This question tests the method of finding real values, examines basic knowledge of circles and line equations, tests reasoning and computational skills, and examines the concepts of functions and equations, reduction and transformation ideas. It is a basic question.

answer:Given: - In triangle ( triangle ABC ), a median ( BM ) is drawn. - ( angle BAC = 30^circ ) - ( angle BMC = 45^circ ) We need to find ( angle BCA ). 1. Since ( BM ) is a median, ( M ) is the midpoint of ( AC ). [ AM = MC ] 2. Given: [ angle BAC = 30^circ ] [ angle BMC = 45^circ ] 3. Considering ( triangle ABM ), we have: [ angle ABM = angle ABC + angle CBM = 45^circ + theta ] Since ( M ) is the midpoint: [ AM = MC Rightarrow triangle AMC text{ isosceles with } angle AMC = angle MCB ] 4. Since the angle sum of ( triangle ABC ) is ( 180^circ ), we have: [ angle BAM + angle BMC + angle BCA = 180^circ ] Substituting the known angles: [ 30^circ + 45^circ + angle BCA = 180^circ ] Solving for ( angle BCA ): [ angle BCA = 180^circ - 30^circ - 45^circ = 105^circ ] 5. Now, considering ( triangle AMB ), since we found that: [ angle ABM = theta text{ and, } 3theta= 105^circ ] 6. Thus, the required angle is: [ boxed{105^circ} ]