answer:Given the vertex of the parabola (2, -3), we know the vertex form of the quadratic equation is: [ y = a(x - 2)^2 - 3 ] This parabola passes through the point (6, -63). Substituting x = 6 and y = -63, we obtain: [ -63 = a(6 - 2)^2 - 3 ] [ -63 = a(16) - 3 ] [ -60 = 16a ] [ a = -60 / 16 ] [ a = -frac{15}{4} ] To verify: Thus, the equation of the parabola is y = -frac{15}{4}(x - 2)^2 - 3. Checking with the original vertex (2, -3): [ y = -frac{15}{4}(2 - 2)^2 - 3 = -3 ] And with point (6, -63): [ y = -frac{15}{4}(6 - 2)^2 - 3 = -frac{15}{4}(16) - 3 = -60 - 3 = -63 ] Everything checks out correctly. Conclusion: The value of a for the given parabola is boxed{-frac{15}{4}}.

answer:Let's denote the side of the square as ( s ), the radius of the circle as ( r ), and the length and breadth of the rectangle as ( l ) and ( b ) respectively. Given that the radius of the circle is equal to the side of the square, we have: [ r = s ] The area of the square is: [ text{Area of square} = s^2 ] The length of the rectangle is one-sixth of the radius of the circle, so: [ l = frac{1}{6}r ] [ l = frac{1}{6}s ] The area of the rectangle is given by: [ text{Area of rectangle} = l times b ] [ 360 = left(frac{1}{6}sright) times 10 ] [ 360 = frac{10}{6}s ] [ 360 = frac{5}{3}s ] [ s = frac{360 times 3}{5} ] [ s = 216 ] Now that we have the side of the square, we can find the area of the square: [ text{Area of square} = s^2 ] [ text{Area of square} = 216^2 ] [ text{Area of square} = 46656 ] Therefore, the area of the square is ( boxed{46656} ) sq. units.

answer:To solve the problem systematically, let's denote the number each student is thinking of as follows: - Let the number student D is thinking of be x. - Since the average of the numbers told by the two people adjacent to A is 1, and one of those numbers is x, the number that student A is thinking of can be expressed as 10-x because the other adjacent number to A would have to balance out x to make the average work out to 1. - Similarly, given that the average announced by C is 3, and one of the adjacent numbers is from A (which we've found to be 10-x), the number that student C is thinking of is x-6. This is because the average of 10-x and x-6 should give 3. - For student E, who announces 5, and knowing that one of the adjacent numbers is x (from D), the number student E is thinking of is 14-x. This ensures that the average of x and 14-x equals 5. - Lastly, considering student B announces 2, and one of the adjacent numbers is from C (x-6), the number that student B is thinking of is x-12. This is because the average of x-6 and another number should result in 2, and that other number, based on the pattern, is x-12. Given these deductions, we can set up an equation based on the information from B and C, since their averages directly involve x: [ x-12 + x = 2 times 3 ] Simplifying this equation: [ 2x - 12 = 6 2x = 18 x = 9 ] Thus, the number that student D is thinking of is 9. Therefore, the correct answer is boxed{D}.

answer:1. **Determine the Area of One Block**: The given problem states that Kolya used four identical blocks to form the figure. Each block has a surface area of 18 cm². 2. **Calculate the Lost Area due to Joins**: When these blocks are joined together, some of the area is "lost" because the surfaces are now in contact with each other. We are told that 2 cm² is lost per block due to these joins. [ text{Effective area of one block} = 18 , text{cm}^2 - 2 , text{cm}^2 = 16 , text{cm}^2 ] 3. **Calculate Total Surface Area for Four Blocks**: Since there are four blocks in total, and each block now effectively contributes 16 cm² to the surface area, the total surface area is: [ text{Total surface area} = 4 times 16 , text{cm}^2 = 64 , text{cm}^2 ] 4. **Conclusion**: The total surface area of the combined figure is: [ boxed{64 , text{cm}^2} ]