Turing

🔍

question:The graph of the linear function y = k(x 1) passes through point M(1, 2). The intersection point of its graph with the yaxis is ( ) A: (0, 1) B: (1, 0) C: (0, 0) D: (0, 1)

answer:Since the graph of the linear function y = k(x - 1) passes through point M(-1, -2), we have k(-1 - 1) = -2, solving this gives k = 1. Therefore, the equation of the function becomes y = x - 1. Substituting x = 0 into this equation gives y = -1. Hence, the intersection point of its graph with the y-axis is (0, -1). Therefore, the correct answer is boxed{A}.

question:Brian’s car gets 20 miles per gallon. On his last trip, he used 3 gallons of gas. He travelled _____ miles on his last trip.

answer:Brian's car gets 20 miles per gallon. If he used 3 gallons of gas on his last trip, then he traveled: 20 miles/gallon * 3 gallons = 60 miles Brian traveled boxed{60} miles on his last trip.

question:is known that triangle ABC is not isosceles with altitudes of AA_1, BB_1 , and CC_1 . Suppose B_A and C_A respectively points on BB_1 and CC_1 so that A_1B_A is perpendicular on BB_1 and A_1C_A is perpendicular on CC_1 . Lines B_AC_A and BC intersect at the point T_A . Define in the same way the points T_B and T_C . Prove that points T_A, T_B , and T_C are collinear.

answer:1. **Define the problem setup:** - Let triangle ABC be a triangle with altitudes AA_1, BB_1, and CC_1. - Points B_A and C_A are on BB_1 and CC_1 respectively such that A_1B_A perp BB_1 and A_1C_A perp CC_1. - Lines B_AC_A and BC intersect at point T_A. - Similarly, define points T_B and T_C. 2. **Lemma:** - Given triangle ABC and a cevian triangle DEF, define X = BC cap EF and similarly Y and Z. - The midpoints of segments AX, BY, and CZ are collinear. 3. **Proof of Lemma:** - This uses the fact that X, Y, and Z are collinear (by the Newton-Gauss Theorem). 4. **Application to the problem:** - Let B_1C_1 cap BC = X and define Y and Z similarly. - We have measuredangle C_1CB = measuredangle BAA_1 = measuredangle C_AA_1H = measuredangle C_AB_AH, implying that ACC_AB_A is cyclic. - Therefore, T_AB cdot T_AC = T_AB_A cdot T_AC_A = TA_1^2, which means that T_A is the midpoint of A_1X. 5. **Conclusion:** - Applying the lemma with respect to triangle A_1B_1C_1, we conclude that the points T_A, T_B, and T_C are collinear. blacksquare

question:a company, 54 percent of the employees are men, while the remaining 46 percent are women. Among the men, 70 percent work fulltime, and 30 percent work parttime. Among the women, 60 percent work fulltime, and 40 percent work parttime. If 60 percent of the fulltime employees are unionized and 50 percent of the parttime employees are unionized, what percent of the nonunion parttime employees are women?

answer:Let's assume there are 100 employees in the company for simplicity. Men: 54% of 100 = 54 men Women: 46% of 100 = 46 women Among the men: Full-time: 70% of 54 = 0.7 * 54 = 37.8 ≈ 38 men (since we can't have a fraction of a person, we'll round to the nearest whole number) Part-time: 30% of 54 = 0.3 * 54 = 16.2 ≈ 16 men Among the women: Full-time: 60% of 46 = 0.6 * 46 = 27.6 ≈ 28 women Part-time: 40% of 46 = 0.4 * 46 = 18.4 ≈ 18 women Total full-time employees: 38 men + 28 women = 66 Total part-time employees: 16 men + 18 women = 34 Unionized full-time employees: 60% of 66 = 0.6 * 66 = 39.6 ≈ 40 (rounding to the nearest whole number) Unionized part-time employees: 50% of 34 = 0.5 * 34 = 17 Non-unionized full-time employees: 66 - 40 = 26 Non-unionized part-time employees: 34 - 17 = 17 Now, we need to find out how many of the non-unionized part-time employees are women. Since 50% of the part-time employees are unionized, the remaining 50% are not. Therefore, the number of non-unionized part-time women is the same as the number of unionized part-time women, which is 50% of the part-time women. Non-unionized part-time women: 50% of 18 = 0.5 * 18 = 9 To find the percent of non-unionized part-time employees who are women, we divide the number of non-unionized part-time women by the total number of non-unionized part-time employees and multiply by 100: Percent of non-unionized part-time employees who are women = (9 / 17) * 100 ≈ 52.94% Therefore, approximately boxed{52.94%} of the non-unionized part-time employees are women.