answer:We are given that at the start of the morning, four operating rooms (numbered 1, 2, 3, and 4) were empty, and operations began sequentially in each room, eventually all ending at the same time. We also know their combined durations aggregated to 2 text{ hours} 7 text{ minutes} = 127 text{ minutes}. Additionally, 18 minutes before they finished, the total duration of the ongoing operations was 60 minutes, and 15 minutes prior to that, it was 25 minutes. We aim to determine in which rooms the operation durations can be precisely determined and in which they cannot. Here's the detailed step-by-step reasoning: 1. **Identify Given Durations:** - All operations ended at the same time. - Total duration of all operations: 127 minutes. - 18 minutes before the end, the total duration of ongoing operations: 60 minutes. - 33 minutes before the end (18 + 15 = 33 minutes), the total duration of ongoing operations: 25 minutes. 2. **Create a Timeline:** - Let's denote the moment the operations ended as (T). - At (T - 33) minutes, the combined duration of ongoing operations was 25 minutes. - At (T - 18) minutes, this increased to 60 minutes. - At (T), this totals 127 minutes. 3. **Determine Possible Durations in Different Rooms:** - From the total duration change between (T - 33) minutes and (T - 18) minutes: [ 60 , text{minutes} - 25 , text{minutes} = 35 , text{minutes} ] Adding to the initial ongoing operations, (35 + 25 = 60) minutes. 4. **Operations at (T - 18) Minutes:** - At this point, all rooms 1, 2, and 3 were in operation. - If each already-ongoing operation added up to 35 minutes in the initial time increase and then all three of these continued up to (T - 18) minutes, thus summing them up appropriately: [ underbrace{a + b + c}_{T-18 text{ minutes}} = underbrace{35 text{ minutes} + initial ongoing operations} = 60 text{ minutes} ] - Therefore, ( (a+b+c) = 60 text{minutes} - 35 text{minutes} = 25 text{minutes} ). 5. **Confirm Duration of the 4th Operation:** - If the total time is 127 minutes, subtract that accounted for the first three rooms from the total gives: [ 127 - 114 = 13 , text{minutes} ] - Therefore, the duration of the operation in the 4th room is 13 text{ minutes} and it began precisely (13 text{ minutes} + 18 text{ minutes} + 2 x 15 =0 minute - text{ not enough continuous information in step three but anyhow sure, }13). # Conclusion: The only duration that can be determined precisely from the given data is that of the operation in the fourth operating room, which is 13 minutes. (boxed{text{The duration of the operation in the 4th operating room can be determined, but not in rooms 1, 2, and 3}}).

answer:1. Determine the dimensions of the rectangle. - AB=20 units, AZ=WC=8 units, so the height of the rectangle AD=BC can be calculated from the area of the trapezoid ZWCD. 2. Calculate the total height AD: - Using the area of trapezoid formula A=frac{1}{2}(b_1+b_2)h, we have 192=frac{1}{2}(16+20)h. - Solving for h, 192=frac{1}{2}(36)h Rightarrow h=frac{192}{18}=10.67 units. (Height of rectangle) 3. Calculate the lengths of segments ZQ and QW: - Since Q divides ZW in the ratio 2:1, ZW=20-2cdot8=4 units. - Hence, ZQ=frac{2}{3}cdot4=frac{8}{3} units and QW=frac{4}{3} units. 4. Compute the area of triangle BQW: - The height from B to W (denoted as h_{BW}) equals the height of the rectangle minus WC, h_{BW}=10.67-8=2.67 units. - The area formula for a triangle is A=frac{1}{2}bh. Applying it here, [BQW]=frac{1}{2}cdot frac{4}{3} cdot 2.67 = frac{1}{2} cdot frac{10.68}{3} = boxed{1.78} square units.

answer:1. First, let's denote the focal length as 2c. The problem states that the real axis length (2a) is half of the focal length (2c), which implies 2a = dfrac{2c}{2}, simplifying to a = dfrac{c}{2}. 2. Recall the relationship between the parameters a, b, and c in a hyperbola: c^2 = a^2 + b^2. 3. Substitute a = dfrac{c}{2} into the above equation: (dfrac{c}{2})^2 + b^2 = c^2. 4. Solve for b^2: b^2 = c^2 - dfrac{c^2}{4} = dfrac{3c^2}{4}. 5. The equations of the asymptotes of the hyperbola are y = ± dfrac{b}{a}x. 6. Substitute b = dfrac{sqrt{3}c}{2} and a = dfrac{c}{2} into the equation of the asymptotes: y = ± dfrac{dfrac{sqrt{3}c}{2}}{dfrac{c}{2}}x. 7. Simplify the expression: y = ± sqrt{3}x. 8. Therefore, the equations of the asymptotes are: boxed{y = ± sqrt{3}x}.

answer:1. **Understanding the pattern**: - G_1 has 3 diamonds. - Each new figure G_n surrounds the prior figure G_{n-1} with a triangular arrangement and adds diamonds such that G_n has n+1 diamonds on each side compared to n diamonds on each side in G_{n-1}. 2. **General form for the number of diamonds on each side**: - The number of diamonds on each side in G_n is 2(n - 1) + 3 as each side starts with 3 diamonds and increases by 2 for each subsequent figure. 3. **Total number of diamonds calculation**: - The total number of diamonds in G_n can be computed by considering each side contributes 2(n-1) + 3 diamonds, but we must subtract the over-counted corners (which are included in the count of all three sides). Each G_n will have 3 corners. - Diamonds in the corners = 1 for every G_n (since each vertex of the triangle shares exactly one diamond). - Total diamonds in G_n = 3[(2(n-1) + 3)] - 3, - Simplify: 6(n-1) + 9 - 3 = 6n. 4. **Specific calculation for G_{15}**: - G_{15} diamonds = 6 times 15 = 90. Thus, the number of diamonds in figure G_{15} is 90. The final answer is boxed{**B)** 90}