answer:Let's calculate the total distance Rich walks step by step. 1. From his house to the sidewalk: 20 feet 2. Down the sidewalk to the end of the road: 200 feet So far, Rich has walked 20 + 200 = 220 feet. 3. He then makes a left and walks double his total distance so far, which is 2 * 220 = 440 feet. Now, the total distance Rich has walked is 220 + 440 = 660 feet. 4. Then he walks half the total distance up to this point again to the end of his route, which is 0.5 * 660 = 330 feet. The total distance Rich has walked to the end of his route is 660 + 330 = 990 feet. Since Rich turns around and walks the same path all the way back home, he will walk another 990 feet. Therefore, the total distance Rich walks is 990 + 990 = boxed{1980} feet.

answer:Solution: (Ⅰ) Since the distance from the moving point M to the line y = -2 is 1 more than its distance to the point F(0, 1), it can be inferred that the distance from the moving point to point F is equal to its distance to the line y = -1, According to the definition of a parabola, the trajectory of the moving point is a parabola with F as the focus and y = -1 as the directrix. Therefore, the equation is x^2 = 4y. ... (4 points) (Ⅱ) Obviously, it is not meaningful if line l is perpendicular to the x-axis, so we can assume the equation of the sought line is y = kx - 1, Substituting into the parabola equation and simplifying, we get: x^2 - 4kx + 4 = 0, ... (6 points) where Delta = 4k^2 + 8 > 0, x_1 + x_2 = -4k, x_1x_2 = 4 ... (8 points) Let point A be (x_1, y_1) and point B be (x_2, y_2), then we have frac{y_1}{x_1} + frac{y_2}{x_2} = 2, ① Since y_1 = kx_1 - 1 and y_2 = kx_2 - 1, substituting into ① and rearranging, we get k = 2, ... (11 points) Therefore, the equation of line l is y = 2x - 1. ... (12 points) Thus, the answers are: - (Ⅰ) The equation of the trajectory of point M is boxed{x^2 = 4y}. - (Ⅱ) The equation of line l is boxed{y = 2x - 1}.

answer:First, let's calculate how much Kathleen saved each month: June: 21 July: 46 August: 45 September: 32 October: Half of what she saved in August, which is 45 / 2 = 22.50 November: Twice the amount she saved in September minus 20, which is (2 * 32) - 20 = 64 - 20 = 44 Now, let's add up all the savings: Total savings = 21 + 46 + 45 + 32 + 22.50 + 44 = 210.50 Next, let's calculate her total expenses: Expenses = 12 (school supplies) + 54 (new clothes) + 37 (birthday gift) + 25 (book) + 10 (donation) = 138 Now, let's subtract her expenses from her total savings: Remaining money = 210.50 - 138 = 72.50 Since Kathleen has saved more than 200, her aunt will give her a bonus of 25. Let's add this to her remaining money: Remaining money with bonus = 72.50 + 25 = 97.50 Therefore, after fulfilling her aunt's conditions and making the donation, Kathleen will have boxed{97.50} left.

answer:Let the weights of the three containers be x, y, and z. From the problem description, the pair weights are: - (x, y) = 234 pounds - (y, z) = 241 pounds - (z, x) = 255 pounds We need to find the total weight x + y + z. Start by summing the pair weights: [ (x+y) + (y+z) + (z+x) = 234 + 241 + 255. ] Calculating the sum: [ 234 + 241 + 255 = 730. ] Each container's weight is counted twice in the sum of all pairings. Therefore, divide the total by 2 to find the combined weight: [ x + y + z = frac{730}{2} = 365. ] Thus, the combined weight in pounds of the three containers is 365. Conclusion: The calculation accurately provides the combined weight, and the problem statement is consistent and solvable. The final answer is boxed{textbf{(B)} 365}