answer:Recall that the diagonals of a rhombus bisect each other at right angles. So, the diagonals divide the rhombus into four right triangles. Each diagonal acts as the hypotenuse of a right triangle, where the half-lengths of the diagonals are the legs. Given diagonals are 16 inches and 30 inches, we calculate the half-lengths of each: - Half of 16 inches is 8 inches. - Half of 30 inches is 15 inches. Using the Pythagorean theorem to find the length of one side of the rhombus: [ text{Side length} = sqrt{8^2 + 15^2} = sqrt{64 + 225} = sqrt{289} = 17 text{ inches} ] The perimeter of the rhombus, being four times the side length, is: [ 4 times 17 = boxed{68} text{ inches} ]

answer:The problem requires calculating the total cost for various ranges of locker numbers with the updated digit cost. 1. **Cost for lockers with 1-digit numbers (1 to 9):** [ text{Cost} = 9 text{ lockers} times 1 text{ digit/locker} times 0.03 text{ dollars/digit} = 0.27 text{ dollars}. ] 2. **Cost for lockers with 2-digit numbers (10 to 99):** [ text{Cost} = 90 text{ lockers} times 2 text{ digits/locker} times 0.03 text{ dollars/digit} = 5.40 text{ dollars}. ] 3. **Cost for lockers with 3-digit numbers (100 to 999):** [ text{Cost} = 900 text{ lockers} times 3 text{ digits/locker} times 0.03 text{ dollars/digit} = 81.00 text{ dollars}. ] 4. **Calculate the cumulative cost to find the target:** [ text{Total cost} = 0.27 + 5.40 + 81.00 = 86.67 text{ dollars}. ] 5. **Remaining cost to hit 206.91:** [ text{Remaining cost} = 206.91 - 86.67 = 120.24 text{ dollars}. ] This corresponds to (frac{120.24 text{ dollars}}{0.03 text{ dollars/digit}} = 4008 text{ digits}). 6. **Determine additional lockers from the remaining digits (4-digits each):** [ text{Additional lockers} = frac{4008}{4} = 1002 text{ lockers}. ] 7. **Calculate the total number of lockers:** [ text{Total lockers} = 999 + 1002 = 2001. ] Thus, the total number of lockers at the school is (2001). Conclusion: The solution is correct based on the new digit cost and total expense calculation. The final answer is boxed{textbf{(A)} 2001}

answer:First, observe that the line from (0,0) to (1432, 876) simplifies as (8k,3k) where 1432 = 179 cdot 8 and 876 = 3 cdot 292. Here, from (8k,3k) to (8(k+1),3(k+1)) represents the basic segment. Translating this line so that (8k,3k) becomes the origin, we check for intersections around (0,0). The new line segment's equation is y = frac{3x}{8}. The points along this segment with integral x-coordinates are (0,0), (1,frac{3}{8}), (2,frac{6}{8}), (3,frac{9}{8}), dots, (8,3). Each segment crossing between (8k,3k) and (8(k+1),3(k+1)) systematically encounters similar square and circle configurations. For a side length frac{1}{4}, the squares extend frac{1}{8} from the center in each direction. Thus, squares centered at lattice points near the origin interfere with the line y=frac{3x}{8} over a range slightly wider than just their center. Likewise, circles with a radius frac{1}{8} also intersect at points each segment. The line segment (8k,3k) to (8(k+1),3(k+1)) intersects 2 squares and 2 circles distinctly, given the setups. Multiplying these intersects over all such segments from (0,0) to (1432,876): For squares and circles: frac{1432}{8} + 1 = 179 + 1 = 180 segments: 180 cdot 2 = 360 squares and 360 circles. Therefore, m + n = 360 + 360 = boxed{720}.

answer:Given that line l is parallel to the line y=-2x+1 and has a y-intercept of -3, we aim to find the equation of line l. 1. Since line l is parallel to y=-2x+1, it must have the same slope. Therefore, the slope of line l is -2. - This gives us the initial form of the equation of line l as y = -2x + b. 2. The y-intercept of a line is the y value when x=0. Given that the y-intercept of line l is -3, we substitute x=0 and y=-3 into the equation. - Substituting these values gives -3 = -2(0) + b. 3. Solving for b, we find that b = -3. - This means the equation of line l is y = -2x - 3. Therefore, the expression of line l in the Cartesian coordinate system is boxed{y = -2x - 3}.