answer:Let us analyze if it's possible to arrange 12 identical coins along the walls of a square box in such a way that a specific number of coins are along each wall. Remember, the coins can be placed on top of one another. 1. **Case a) Exactly 2 coins along each wall:** Each wall needs 2 coins. Since the box has 4 walls, the total number of coins needed is: [ 2 times 4 = 8 ] Since 8 coins are required but only 12 coins are provided, it is not possible to place exactly 2 coins along each wall. Therefore, this case is not possible. [ boxed{a: text{no}} ] 2. **Case b) Exactly 3 coins along each wall:** Similarly, each wall needs 3 coins. Thus, the total number of coins needed is: [ 3 times 4 = 12 ] In this case, it is possible to place 3 coins along each wall since we have exactly 12 coins. The arrangement can be visualized by placing 3 coins per wall. [ boxed{b: text{yes}} ] 3. **Case c) Exactly 4 coins along each wall:** Each wall needs 4 coins. Therefore, the total number of coins required is: [ 4 times 4 = 16 ] We need 16 coins, but only 12 coins are available, so it is not possible to place 4 coins along each wall. [ boxed{c: text{no}} ] 4. **Case d) Exactly 5 coins along each wall:** Each wall needs 5 coins. Thus, the total number of coins required is: [ 5 times 4 = 20 ] We need 20 coins, but only 12 coins are available, so it is not possible to place 5 coins along each wall. [ boxed{d: text{no}} ] 5. **Case e) Exactly 6 coins along each wall:** Each wall needs 6 coins. Therefore, the total number of coins required is: [ 6 times 4 = 24 ] We need 24 coins, but only 12 coins are available, so it is not possible to place 6 coins along each wall. [ boxed{e: text{no}} ] # Conclusion: - Placing 2 coins along each wall: boxed{text{a: no}} - Placing 3 coins along each wall: boxed{text{b: yes}} - Placing 4 coins along each wall: boxed{text{c: no}} - Placing 5 coins along each wall: boxed{text{d: no}} - Placing 6 coins along each wall: boxed{text{e: no}} Note: There was only one correct case (`b`), so it's not useful for options (f) since there was no case specified for it so it was considered as NO.

answer:Let E be the midpoint of overline{QR}. By SAS Congruence, triangle PQE cong triangle PRE, so angle PQE = angle PRE = 90^circ. Let QE = z, PQ = w, and angle QJE = phi, where phi = frac{angle PQE}{2}. Then cos(phi) = frac{z}{10} and cos(2phi) = frac{z}{w} = 2cos^2(phi) - 1 = frac{z^2 - 50}{50}. Cross-multiplying gives 50z = w(z^2 - 50). Since w,z > 0, z^2 - 50 must be positive, implying z > sqrt{50}. Also, since triangle QJE has hypotenuse overline{QJ} of length 10, QE = z < 10. Given that QR = 2z is an integer, possible values for z are 8, 9, and 9.5. Only z = 8 and z = 9 yield integer values for PQ = w. Calculating for each: - For z = 8, w = frac{50 times 8}{8^2 - 50} = 100. - For z = 9, w = frac{50 times 9}{9^2 - 50} = 90. Thus, the perimeter of triangle PQR for z = 9 and w = 90 is 2(w + z) = 2(90 + 9) = boxed{198}.

answer:1. **Identify the radii and areas of each circle**: - Circle 1: Radius = 3 inches, Area = 9pi square inches. - Circle 2: Radius = 6 inches, Area = 36pi square inches. - Circle 3: Radius = 9 inches, Area = 81pi square inches. - Circle 4: Radius = 12 inches, Area = 144pi square inches. - Circle 5: Radius = 15 inches, Area = 225pi square inches. - Circle 6: Radius = 18 inches, Area = 324pi square inches. - Circle 7: Radius = 21 inches, Area = 441pi square inches. 2. **Calculate the total area of the largest circle**: - Total area = 441pi square inches. 3. **Determine the areas of the black regions**: - Black circle areas: - Circle 1: 9pi square inches. - Circle 3: 81pi - 36pi = 45pi square inches. - Circle 5: 225pi - 144pi = 81pi square inches. - Circle 7: 441pi - 324pi = 117pi square inches. - Total area of black regions = 9pi + 45pi + 81pi + 117pi = 252pi square inches. 4. **Calculate the percentage of the design that is black**: [ text{Percentage} = left(frac{252pi}{441pi}right) times 100 = frac{252}{441} times 100 approx 57.14% ] 5. **Conclusion**: The closest percent of the design that is black is approximately 57%. [ 57% ] The final answer is boxed{(C) 57%}

answer:Reflecting a point over the x-axis changes the sign of the y-coordinate, keeping the x-coordinate unchanged. Thus: - For vertex E(2, 3), its reflection E' over the x-axis will have coordinates E'(2, -3). - To find the distance between E and E', we use the distance formula for points in a plane: [ text{Distance} = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] where (x_1, y_1) = (2, 3) and (x_2, y_2) = (2, -3): [ text{Distance} = sqrt{(2 - 2)^2 + (-3 - 3)^2} = sqrt{0 + (-6)^2} = sqrt{36} = boxed{6} ]