answer:Tom mows 3 lawns and charges 12 per lawn. So, the total amount he earns from mowing lawns is: 3 lawns * 12/lawn = 36 He also made extra money pulling weeds for 10. So, the total amount he earned last month is: 36 (from mowing) + 10 (from weeding) = 46 Tom spends 17 on gas each month. To find his profit, we subtract his expenses from his earnings: 46 (total earnings) - 17 (gas expenses) = 29 Tom's profit last month was boxed{29} .

answer:When the ball is dropped from a height h, it will first fall a distance of h to the ground. After that, it will bounce back up to a height of 0.8h (80% of the original height), then fall back down the same distance of 0.8h. This process will continue indefinitely, with the ball reaching 80% of the previous height on each bounce. To calculate the total distance covered by the ball, we need to sum the distances of all the bounces. The total distance D can be expressed as an infinite geometric series: D = h + 2(0.8h) + 2(0.8^2h) + 2(0.8^3h) + ... This is a geometric series with the first term a = h and the common ratio r = 2 * 0.8 = 1.6. However, since the ball is not bouncing back to the full height, we need to adjust the common ratio for the subsequent terms after the initial drop. The correct common ratio for the bounce distances is r = 0.8. So the series becomes: D = h + 2(0.8h) + 2(0.8^2h) + 2(0.8^3h) + ... To find the sum of this infinite geometric series, we use the formula for the sum of an infinite geometric series, which is: S = a / (1 - r) where S is the sum of the series, a is the first term, and r is the common ratio. In this case, the first term a is not just h, but h + 2(0.8h), because the first bounce up and down is included in the first term. So we have: a = h + 2(0.8h) = h + 1.6h = 2.6h The common ratio r is 0.8 (for the bounce distances). Now we can find the sum of the series: S = a / (1 - r) S = 2.6h / (1 - 0.8) S = 2.6h / 0.2 S = 13h So the total distance covered by the ball is boxed{13} times the original height h from which it was dropped.

answer:1. **Color Grid Like a Checkerboard:** - Color the grid such that every other square is black or white, similar to an infinite checkerboard. - We start by fixing the square at ( (0,0) ) as a black square. - Therefore, the square ((x, y)) is black if (x equiv y pmod{2}) is true. Otherwise, it is white. 2. **Identify Initial and Goal Colors:** - The werewolf starts at position ((26, 10)). - Calculate the parity of (26) and (10): [ 26 + 10 = 36 quad text{(even)} ] Since (26 + 10 equiv 0 pmod{2}), the initial position ((26, 10)) is black. - The goal is to reach position ((42, 2017)): - Calculate the parity of (42) and (2017): [ 42 + 2017 = 2059 quad text{(odd)} ] Since (42 + 2017 equiv 1 pmod{2}), the goal position ((42, 2017)) is white. 3. **Analyze Movement Constraints:** - The given allowable movements are ({(3, 2), (2, 3), (-2, 3), (-3, 2), (-3, -2), (-2, -3), (2, -3), (3, -2)}). - Notice that each movement changes the color of the current square: - For example, if at position ((x, y)), after moving ( (3, 2) ): [ x+3 quad text{and} quad y+2 implies quad (x+3 + y+2) = (x + y + 5) ] Which means the parity of the sum of the coordinates changes (odd to even or even to odd). 4. **Conclude Feasibility of Reaching Goals:** - In each move, the sum of the coordinates changes parity. - Since the initial position ((26, 10)) has an even sum and the goal position ((42, 2017)) has an odd sum, the werewolf can only remain on squares that maintain the same color as the initial square (black). - Therefore, it is impossible for the werewolf to reach a goal on a square with a different color (white). 5. **Final Conclusion:** - Given the constraints and color parity analysis, we conclude that the werewolf can never reach the goal position located at ((42, 2017)). Thus, [boxed{text{It is not possible for the werewolf to reach the goal position.}}]

answer:To find all ways to cut the plywood into 6 congruent rectangles, consider valid dimensions for the rectangles formed: 1. If all cuts are parallel to the 9-foot side, each rectangle would measure frac{6}{3} times 9 = 2 times 9. The perimeter is then 2(2 + 9) = 22 feet. 2. If all cuts are parallel to the 6-foot side, each rectangle would be frac{9}{3} times 6 = 3 times 6. The perimeter of each would be 2(3 + 6) = 18 feet. 3. If three cuts are parallel to the 9-foot side and two cuts are parallel to the 6-foot side, the resulting rectangles have dimensions frac{6}{2} times frac{9}{3} = 3 times 3. The perimeter is 2(3 + 3) = 12 feet. 4. If two cuts are parallel to the 9-foot side and three cuts are parallel to the 6-foot side, the rectangles measure frac{6}{3} times frac{9}{2} = 2 times 4.5. The perimeter would be 2(2 + 4.5) = 13 feet. The differences between the greatest and least perimeters are 22 - 12 = boxed{10} feet.