answer:1. **Assume the total capacity of the pitcher**: Let's denote the total capacity of the pitcher as C ounces. 2. **Calculate the amount of juice in the pitcher**: Since the pitcher is filled to three-fourths of its capacity with pineapple juice, the amount of juice in the pitcher is frac{3}{4}C ounces. 3. **Determine the amount of juice per cup**: The juice is evenly distributed into 5 cups. Therefore, the amount of juice in each cup is: [ frac{frac{3}{4}C}{5} = frac{3C}{20} text{ ounces} ] 4. **Calculate the percentage of the total capacity per cup**: To find out what percentage of the total pitcher capacity each cup received, we calculate: [ left(frac{frac{3C}{20}}{C}right) times 100% = frac{3C}{20C} times 100% = frac{3}{20} times 100% = 15% ] 5. **Conclusion**: Each cup received 15% of the total capacity of the pitcher. Thus, the correct answer is boxed{textbf{(C) }15}.

answer:**Answer**: (1) From the given information, we know that y' = 2x + 1. The slope of line l_1 is k = 2 times 1 + 1 = 3, so the equation of line l_1 is y = 3x - 3. Let line l_2 pass through point B(b, b^2 + b - 2) on the curve y = x^2 + x - 2, then the equation of l_2 is y = (2b + 1)x - b^2 - b - 2. Since l_1 perpendicular l_2, we have 2b + 1 = -frac{1}{3}, thus b = -frac{2}{3}, and hence the equation of l_2 is y = -frac{1}{3}x - frac{2}{3}. (2) The intersection point of l_1 and l_2 is (x, y), and the intersection points of l_1, l_2 with the x-axis are (1, 0) and (-frac{2}{3}, 0), respectively. Therefore, the area S of the triangle is S = frac{1}{2} times text{base} times text{height} = boxed{text{answer}}.

answer:To prove that the number of outside entrance doors is even, given that every room has an even number of doors leaving it, we will approach the proof step-by-step. 1. **Definition and Initial Observations**: - Let's denote the total number of rooms as (R). - Let the total number of doors be (D). - Each door connects either: - One room to another room, or - One room to the outside. 2. **The Double-Counting Principle**: - Each door can be considered to have two "sides." - If a door connects room (i) to room (j), then both (i) and (j) have a door leaving them. - If a door connects room (i) to the outside, then room (i) has a door leaving it to the outside. - Consequently, whether a door is between two rooms or between a room and the outside, it will always have two sides contributing to the total count. 3. **Counting the Total Number of Door Sides**: - Each door has 2 sides. - Therefore, the total number of door sides is (2D). 4. **Sum of Sides Per Room**: - Given that every room has an even number of doors, the number of sides leaving each room is even. - Let's denote the number of doors leaving room (i) as (d_i). - Since every (d_i) is even, their contribution to the total number of door sides from all rooms combined is: [ sum_{i=1}^{R} d_i ] - This sum is even because it is the sum of even numbers. 5. **Contribution of Outside Entrances**: - Let the number of doors connecting directly to the outside be (E). - The sum of all sides must account for doors directly connecting to the outside as their sides add up similarly. - Each entrance to the outside also contributes exactly 1 additional side pointing to the outside part, thus (E) additional sides. 6. **Evenness Argument**: - The total number of door sides can be written as: [ 2D = left(sum_{i=1}^{R} d_iright) + E ] - The left-hand side ((2D)) is obviously even, as it is twice the total number of doors. - The first term on the right-hand side, (sum_{i=1}^{R} d_i), is even because it is a sum of even numbers. - For the whole expression to maintain its evenness, (E) must also be even. # Conclusion: [ boxed{text{The number of outside entrance doors, denoted by } E, text{ is even.}} ]

answer:Let's denote the total number of voters in county X as 2V and in county Y as V, according to the ratio 2:1. Let's assume that Douglas won P percent of the vote in county X. The total number of votes Douglas got in county X is then (P/100) * 2V. The total number of votes Douglas got in county Y is (46/100) * V. According to the information given, Douglas won 58 percent of the total votes in counties X and Y combined. The total number of votes in counties X and Y is 2V + V = 3V. The total number of votes Douglas got in counties X and Y is (58/100) * 3V. Now we can set up the equation: (P/100) * 2V + (46/100) * V = (58/100) * 3V Solving for P: (2PV + 46V) / 100 = (174V) / 100 2PV + 46V = 174V 2PV = 174V - 46V 2PV = 128V P = 128V / 2V P = 64 Therefore, candidate Douglas won boxed{64} percent of the vote in county X.