answer:To calculate the total number of bottles of champagne Ashley needs for her wedding, we follow these steps: 1. **Calculate the total number of servings needed:** Ashley wants to serve 2 glasses of champagne to each of her 120 wedding guests. Therefore, the total number of servings needed is calculated as follows: [ text{Total servings} = 2 times 120 = 240 ] 2. **Determine the number of bottles required:** Since each bottle of champagne has 6 servings, the number of bottles needed to serve 240 servings is calculated by dividing the total servings by the servings per bottle: [ text{Number of bottles} = frac{240}{6} = 40 ] Therefore, Ashley will need boxed{40} bottles of champagne for her wedding toast.

answer:- The smallest three-digit number is 100. - To find the smallest three-digit multiple of 13, we compute the ceiling of 100 divided by 13 and then multiply by 13. [ text{Smallest multiple} = leftlceil frac{100}{13} rightrceil times 13 = 8 times 13 = 104 ] Therefore, the smallest three-digit multiple of 13 is boxed{104}.

answer:Step 1: Compute distances between pairs of points. [ AB = sqrt{(2-1)^2 + (0-4)^2 + (-7-2)^2} = sqrt{1 + 16 + 81} = sqrt{98} ] [ AC = sqrt{(5-1)^2 + (-5-4)^2 + (1-2)^2} = sqrt{16 + 81 + 1} = sqrt{98} ] [ BC = sqrt{(5-2)^2 + (-5-0)^2 + (1+7)^2} = sqrt{9 + 25 + 64} = sqrt{98} ] Step 2: Since ( AB = AC = BC = sqrt{98} ), the triangle ( ABC ) is equilateral. Step 3: If ( s ) is the side length of the cube, then ( ssqrt{2} = sqrt{98} ) (since the diagonal of a square face of a cube is ( ssqrt{2} )). [ s = frac{sqrt{98}}{sqrt{2}} = sqrt{49} = 7 ] Step 4: Compute the surface area of the cube. [ text{Surface Area} = 6s^2 = 6 times 7^2 = 6 times 49 = 294 ] [ boxed{294} ] Conclusion: The surface area of the cube is boxed{294}.

answer:To find f(-2), we consider the given functional relationship f(3x+1). We need to figure out the value of x which makes the argument 3x+1 equal to -2. Hence, we set up the equation: [ 3x+1 = -2 ] We solve for x: begin{align*} 3x + 1 &= -2 3x &= -2 - 1 3x &= -3 x &= -1 end{align*} Now that we have found x = -1, we can substitute this value into the given function to find f(-2): begin{align*} f(-2) &= f(3(-1) + 1) &= f(3 cdot (-1) + 1) &= f(0) &= 9(-1)^2 - 6(-1) + 5 &= 9 cdot 1 + 6 + 5 &= 9 + 6 + 5 &= 20 end{align*} Therefore, f(-2) = boxed{20}.