💡First, I need to determine the size of the smaller pack of soymilk that Hannah bought. I know that soymilk is sold in packages of two sizes: a smaller pack and a larger pack. Hannah purchased a total of 17 soymilk cartons, with 3 of them being large packs.Let's denote the size of the smaller pack as ( S ) and the size of the larger pack as ( L ). The total number of soymilk cartons from the large packs is ( 3L ). Therefore, the remaining number of soymilk cartons from the smaller packs is ( 17 - 3L ).Since the smaller packs contain exactly 2 cartons each, the remaining number of cartons must be a multiple of 2. This means that ( 17 - 3L ) must be an even number.To find a suitable value for ( L ), I'll start by testing common sizes for larger packs. Let's try ( L = 5 ):[17 - 3 times 5 = 17 - 15 = 2]This leaves 2 soymilk cartons, which is an even number and fits the size of the smaller pack. Therefore, the size of the smaller pack is 2.

💡Alright, so I have this problem here about three sectors arranged to form a closed figure, and I need to find its perimeter. Each sector has a radius of 100 cm and a central angle of 60 degrees. Hmm, okay, let me try to visualize this. If each sector is like a slice of a circle with a 60-degree angle, then three of them would make up a full circle, right? Because 60 times 3 is 180, wait, no, 60 times 3 is 180? Wait, no, 60 times 3 is actually 180? Wait, no, 60 times 3 is 180? Wait, no, that's not right. 60 times 3 is 180? Wait, no, 60 times 3 is 180? Wait, no, 60 times 3 is 180? Wait, hold on, 60 times 3 is actually 180? No, wait, 60 times 3 is 180? Wait, 60 times 3 is 180? Wait, no, 60 times 3 is 180? Wait, no, 60 times 3 is 180? Wait, I'm getting confused here. Let me think again. 60 degrees times 3 would be 180 degrees, which is half a circle, not a full circle. So, three sectors of 60 degrees each would make up a semicircle? Hmm, but the problem says they're arranged to form a closed figure. Maybe it's not a semicircle then. Maybe it's something else.Wait, perhaps it's a Reuleaux triangle? I remember that shape, it's like a curved triangle formed by three circular arcs. Each arc is a 60-degree sector of a circle. So, if each sector has a radius of 100 cm, then the perimeter of the Reuleaux triangle would be three times the length of one arc. That makes sense because each side of the Reuleaux triangle is an arc of 60 degrees.Okay, so to find the perimeter, I need to calculate the length of one arc and then multiply it by three. The formula for the length of an arc is (θ/360) times the circumference of the full circle. The circumference of a full circle is 2πr, so for one arc, it would be (60/360) * 2πr. Simplifying that, 60/360 is 1/6, so the arc length is (1/6) * 2πr, which is (1/3)πr.Given that the radius r is 100 cm, plugging that in, the arc length is (1/3) * π * 100. So that's (100/3)π cm. Since we have three such arcs, the total perimeter would be 3 * (100/3)π, which simplifies to 100π cm.Wait, but the problem says to use π = 3.14. So, substituting that in, the perimeter is 100 * 3.14, which is 314 cm. Hmm, that seems straightforward. But let me double-check to make sure I didn't make any mistakes.First, confirming the arc length formula: yes, arc length is (θ/360) * 2πr. For θ = 60 degrees, that's (60/360) = 1/6. So, 1/6 * 2πr = (1/3)πr. That's correct. Then, multiplying by 3 gives 100π, which is 314 cm when π is 3.14. Yeah, that seems right.But just to be thorough, let me think about the shape again. If it's a Reuleaux triangle, it's a curve of constant width, and each arc is indeed 60 degrees. So, the perimeter is just three times the arc length. So, I think my calculation is correct.Wait, another way to think about it: the circumference of a full circle with radius 100 cm is 2π*100 = 200π cm. Since each sector is 60 degrees, which is 1/6 of a full circle, the arc length is 1/6 of 200π, which is (200π)/6 = (100π)/3 cm. Then, three such arcs would be 3*(100π)/3 = 100π cm. Yep, same result. So, that's consistent.Therefore, using π = 3.14, the perimeter is 100 * 3.14 = 314 cm. I think that's the answer.But just to make sure, let me consider if there's any other part of the shape that contributes to the perimeter. If the sectors are arranged such that their straight sides are connected, then those straight sides would form an equilateral triangle, right? Because each central angle is 60 degrees, so the angle between the radii is 60 degrees, making the triangle equilateral. However, in a Reuleaux triangle, the perimeter is only the three arcs, not including the straight sides. So, in this case, since the problem mentions the perimeter of the closed figure, and the closed figure is formed by the three arcs, the straight sides are internal and not part of the perimeter. Therefore, we only need to consider the three arcs for the perimeter.So, confirming again, the perimeter is 314 cm. I think that's solid.**Final Answer**The perimeter of the closed figure is boxed{314} centimeters.

💡First, I need to determine the increase in average weight when a new person replaces one of the original four persons. The original person weighs 95 kg, and the new person weighs 129 kg.I'll start by calculating the increase in total weight by subtracting the weight of the person who left from the weight of the new person:129 kg - 95 kg = 34 kgNext, to find the increase in average weight, I'll divide the total increase in weight by the number of persons, which is 4:34 kg / 4 = 8.5 kgTherefore, the average weight increased by 8.5 kg.

💡First, I need to determine the percentage of students who failed in both Hindi and English. I know that 34% failed in Hindi, 44% failed in English, and 44% passed in both subjects.To find the percentage who failed in at least one subject, I subtract the percentage who passed in both from the total percentage of students:100% - 44% = 56%.Using the principle of inclusion-exclusion, the percentage who failed in at least one subject is the sum of those who failed in Hindi and English minus those who failed in both:34% + 44% - x% = 78% - x%.Setting this equal to the previously calculated 56%:78% - x% = 56%.Solving for x gives:x% = 78% - 56% = 22%.Therefore, 22% of the students failed in both Hindi and English.

💡First, I need to find the complement of the given angle α, which is 63 degrees and 21 minutes. Complementary angles add up to 90 degrees.To subtract the given angle from 90 degrees, I'll start by subtracting the degrees: 90 degrees minus 63 degrees equals 27 degrees.Next, I'll subtract the minutes: 0 minutes minus 21 minutes. Since I can't subtract 21 minutes from 0 minutes, I'll borrow 1 degree from the degrees, which is equivalent to 60 minutes. This leaves me with 26 degrees and 60 minutes.Now, I can subtract the minutes: 60 minutes minus 21 minutes equals 39 minutes.Combining the degrees and minutes, the complement of angle α is 26 degrees and 39 minutes.

💡First, I will start by examining the given equation ( x^2 - 10x = 39 ). To find the positive solution, I can complete the square. I will add ( 25 ) to both sides of the equation to complete the square:[x^2 - 10x + 25 = 39 + 25]This simplifies to:[(x - 5)^2 = 64]Taking the square root of both sides gives:[x - 5 = pm 8]Solving for ( x ), I get two solutions:[x = 5 + 8 = 13 quad text{and} quad x = 5 - 8 = -3]Since we are looking for the positive solution, ( x = 13 ).Next, I need to express ( x ) in the form ( sqrt{a} - b ). From the equation ( (x - 5)^2 = 64 ), I can rewrite ( x ) as:[x = sqrt{64} + 5 = 8 + 5 = 13]This means ( a = 64 ) and ( b = 5 ).Finally, adding ( a ) and ( b ) together:[a + b = 64 + 5 = 69]