answer:Let's denote the cost of the bench as B dollars. According to the problem, the garden table costs 2 times the price of the bench, so the cost of the garden table is 2B dollars. The combined cost of the garden table and the bench is given as 450 dollars. Therefore, we can write the equation: B (cost of the bench) + 2B (cost of the garden table) = 450 dollars Combining like terms, we get: 3B = 450 dollars To find the cost of the bench, we divide both sides of the equation by 3: B = 450 dollars / 3 B = 150 dollars So, the cost of the bench is boxed{150} dollars.

answer:1. **Identify the problem**: We need to find the number of Hamiltonian cycles in a cube graph, where each vertex represents a friend, and each edge represents a possible path between friends. A Hamiltonian cycle visits each vertex exactly once and returns to the starting vertex. 2. **Vertices and edges of a cube**: A cube has 8 vertices and 12 edges. Each vertex is connected to 3 other vertices. 3. **Starting at any vertex**: We can start at any of the 8 vertices. 4. **First move**: From the starting vertex, we have 3 possible edges to choose from. 5. **Second move**: From the second vertex, we have 2 possible edges to choose from (since we cannot go back to the starting vertex). 6. **Third move**: From the third vertex, we again have 2 possible edges to choose from (since we cannot go back to the previous vertex). 7. **Fourth move**: From the fourth vertex, there is only 1 viable path that continues the cycle without revisiting any vertex. 8. **Calculation**: The number of different paths can be calculated as follows: [ 8 text{ (starting vertices)} times 3 text{ (first move)} times 2 text{ (second move)} times 2 text{ (third move)} = 96 ] Thus, the number of different Hamiltonian cycles in the cube is ( boxed{96} ).

answer:1. Given the functional equation ( f(x_1 x_2) = f(x_1) + f(x_2) ) for any positive numbers ( x_1 ) and ( x_2 ), along with the specific value ( f(8) = 3 ). 2. To find ( f(2) ): - By applying the given functional equation: [ f(2 cdot 2) = f(2) + f(2) ] - This simplifies to: [ f(4) = 2f(2) ] - Apply the functional equation again: [ f(4 cdot 2) = f(4) + f(2) ] - This simplifies to: [ f(8) = f(4) + f(2) ] - Given ( f(8) = 3 ) and substituting ( f(4) = 2f(2) ): [ 3 = 2f(2) + f(2) implies 3 = 3f(2) ] - Solving for ( f(2) ): [ f(2) = 1 ] 3. To find ( f(sqrt{2}) ): - By applying the functional equation: [ f(sqrt{2} cdot sqrt{2}) = f(sqrt{2}) + f(sqrt{2}) ] - This simplifies to: [ f(2) = 2f(sqrt{2}) ] - Given ( f(2) = 1 ): [ 1 = 2f(sqrt{2}) ] - Solving for ( f(sqrt{2}) ): [ f(sqrt{2}) = frac{1}{2} ] 4. Conclusion: [ boxed{ frac{1}{2} } ]

answer:1. **Determine the diameter of each salami circle**: Given that eight salami circles fit exactly across the diameter of a 16-inch plate: [ text{Diameter of each salami} = frac{16 text{ inches}}{8} = 2 text{ inches} ] 2. **Calculate the radius of each salami circle**: The radius is half of the diameter: [ text{Radius of each salami} = frac{2}{2} = 1 text{ inch} ] 3. **Calculate the area of one salami circle**: Using the formula for the area of a circle, (A = pi r^2): [ text{Area of one salami} = pi times (1)^2 = pi text{ square inches} ] 4. **Calculate the total area covered by 32 salami circles**: [ text{Total area covered by salami} = 32 times pi = 32pi text{ square inches} ] 5. **Calculate the area of the plate**: The radius of the plate is half its diameter (8 inches). Thus, the area of the plate is: [ text{Area of the plate} = pi times (8)^2 = 64pi text{ square inches} ] 6. **Calculate the fraction of the plate covered by salami**: The fraction covered by salami is the ratio of the total area covered by the salami to the total area of the plate: [ text{Fraction covered} = frac{32pi}{64pi} = frac{32}{64} = frac{1}{2} ] 7. **Conclusion**: The fraction of the plate that is covered by salami is (frac{1}{2}). Therefore, the correct answer is: [ frac{1{2}} ] The final answer is boxed{textbf{(C)} frac{1}{2}}