answer:1. Denote the plane of triangle ABC by Sigma. According to the problem, SD perp Sigma. 2. Therefore, triangle ADS is a right triangle, with the angle at A equal to the angle between SA and Sigma, which is 45^circ. 3. Since triangle ADS is right and isosceles with AS = 2a, we have AD = DS = asqrt{2}. 4. Points B and C lie on the circle k centered at A with radius a in Sigma, and they also lie on the sphere g centered at S with radius asqrt{3}. 5. Since SD = asqrt{2} < asqrt{3}, the sphere g intersects Sigma in a circle k_1 centered at D with radius sqrt{(asqrt{3})^2 - (asqrt{2})^2} = a. 6. The square AB^{*}D C^{*} has AD as one of its diagonals. The vertices B^{*} and C^{*} on the circles k and k_1. Since there can be at most two intersection points of two circles, B and C (in some order) coincide with B^{*} and C^{*}, respectively. Thus, ABDC is a square. 7. Let O be the center of square ABDC. Using the Pythagorean theorem in the triangle ODS, we have: [ OS^2 = OD^2 + DS^2 ] Using OD = frac{1}{2} sqrt{2} a = asqrt{frac{1}{2}}, this gives: [ OS^2 = left(asqrt{frac{1}{2}}right)^2 + (asqrt{2})^2 = frac{a^2}{2} + 2a^2 = frac{5}{2} a^2 ] 8. To find the surface area of the tetrahedron SABC, we find the areas of triangles SBC, SAB, SAC, and ABC. - The area of triangle SBC: [ text{Area}_{SBC} = frac{1}{2} times BS times SC times sin(angle BSC) = frac{1}{2} times asqrt{3} times a sqrt{3} times frac{sqrt{2}}{2} = frac{1}{2} times 3a^2 times frac{sqrt{2}}{2} = frac{3sqrt{2}a^2}{2} ] - The areas of triangles SAB and SAC are equal, and using side lengths a, a sqrt{3}, and 2a, we can use Heron’s formula: [ text{Area}_{SAB} = text{Area}_{SAC} = frac{sqrt{3}}{4}(2a)^2 = a^2 sqrt{3} ] - The area of triangle ABC: [ text{Area}_{ABC} = frac{sqrt{3}}{4}a^2 ] 9. Finally, we add up these areas to find the surface area of the tetrahedron: [ text{Surface area} = text{Area}_{SBC} + 2 times text{Area}_{SAB} + text{Area}_{ABC} ] [ = frac{3sqrt{2}a^2}{2} + 2a^2 sqrt{3} + frac{sqrt{3}}{4}a^2 ] [ = frac{3sqrt{2}a^2 + 8sqrt{3}a^2 + sqrt{3}a^2/4}{2} ] [ = frac{3a^2(sqrt{2} + sqrt{3} + sqrt{5})}{2} ] 10. Thus, the surface area of the tetrahedron SABC is: [ boxed{frac{a^2}{2}(1+sqrt{3}+sqrt{5})} ]

answer:Given 4^{x} = 9^{y} = 6, we can express x and y in terms of logarithms as follows: Since 4^{x} = 6, we get x = frac{log 6}{log 4}. Analogously, since 9^{y} = 6, we obtain y = frac{log 6}{log 9}. Now, let's compute the required sum: begin{align*} frac{1}{x} + frac{1}{y} &= frac{log 4}{log 6} + frac{log 9}{log 6} &= frac{log 2^2}{log 6} + frac{log 3^2}{log 6} &= frac{2log 2}{log 6} + frac{2log 3}{log 6} &= 2 left( frac{log 2 + log 3}{log 6} right) &= 2 left( frac{log (2 cdot 3)}{log 6} right) &= 2 left( frac{log 6}{log 6} right) &= 2 cdot 1 &= boxed{2}. end{align*}

answer:**Step 1: Calculate the side length and height of the outer triangle.** Given the area of an equilateral triangle: [ A = frac{s^2 sqrt{3}}{4} ] Where [ s = sqrt{frac{4A}{sqrt{3}}} = sqrt{frac{4 times 36}{sqrt{3}}} = 12 ] Then, the height ( h ) of the outer triangle: [ h = sqrt{3}frac{s}{2} = sqrt{3} times 6 = 6sqrt{3} ] **Step 2: Determine side length and area of the inner triangle using its height.** The height ( h_{text{inner}} ) of the inner triangle is ( frac{2}{3} ) of ( h ): [ h_{text{inner}} = frac{2}{3} times 6sqrt{3} = 4sqrt{3} ] Thus, the side length ( s_{text{inner}} ) of the inner triangle is: [ s_{text{inner}} = frac{2 h_{text{inner}}}{sqrt{3}} = frac{2 times 4sqrt{3}}{sqrt{3}} = 8 ] [ text{Area of the inner equilateral triangle} = frac{s_{text{inner}}^2 sqrt{3}}{4} = frac{64 times sqrt{3}}{4} = 16sqrt{3} ] **Step 3: Calculate total area between triangles and area of one trapezoid.** [ text{Area difference} = 36 - 16sqrt{3} ] [ text{Area of one trapezoid} = frac{36 - 16sqrt{3}}{3} ] **Conclusion:** Each trapezoid has an area of [ frac{36 - 16sqrt{3}{3}} ] boxed{The correct answer is (textbf{(E)} frac{36 - 16sqrt{3}}{3}).}

answer:Ricky originally had 40 roses. His sister stole 4 roses, leaving him with: 40 - 4 = 36 roses He wants to give away these 36 roses in equal portions of 4 roses per person. To find out how many people will receive the roses, we divide the remaining number of roses by the number of roses each person gets: 36 roses ÷ 4 roses/person = 9 people So, boxed{9} people will receive the roses.