answer：Alright, so I have this problem here: Given the function ( f(x) = x sin x ), where ( x ) is a real number, I need to figure out the relationship among ( fleft( frac{pi}{5} right) ), ( f(1) ), and ( fleft( -frac{pi}{3} right) ). The options are A, B, C, and D, each giving a different order of these function values.First, let me understand the function ( f(x) = x sin x ). It's a product of ( x ) and ( sin x ). I know that ( sin x ) is an odd function, meaning ( sin(-x) = -sin x ), and ( x ) is also an odd function because ( -x = -1 cdot x ). So, when I multiply two odd functions, I get an even function because ( (-x) sin(-x) = (-x)(-sin x) = x sin x ). Therefore, ( f(x) ) is an even function. That means ( f(-x) = f(x) ). So, ( fleft( -frac{pi}{3} right) = fleft( frac{pi}{3} right) ).Okay, so now I can rewrite the problem in terms of positive arguments:- ( fleft( frac{pi}{5} right) )- ( f(1) )- ( fleft( frac{pi}{3} right) )So, I need to compare these three values.Next, I should figure out the behavior of the function ( f(x) = x sin x ). Maybe I can find its derivative to see where it's increasing or decreasing.Let's compute the derivative ( f'(x) ):( f'(x) = frac{d}{dx} [x sin x] = sin x + x cos x ).So, ( f'(x) = sin x + x cos x ).Now, I need to analyze the sign of ( f'(x) ) in the interval ( (0, frac{pi}{2}) ) because all my arguments ( frac{pi}{5} ), ( 1 ), and ( frac{pi}{3} ) lie within this interval.Let me evaluate ( f'(x) ) in this interval:1. ( sin x ) is positive in ( (0, pi) ), so ( sin x > 0 ).2. ( cos x ) is positive in ( (0, frac{pi}{2}) ), so ( x cos x > 0 ).Therefore, both terms ( sin x ) and ( x cos x ) are positive in ( (0, frac{pi}{2}) ), which means ( f'(x) > 0 ) in this interval. So, ( f(x) ) is increasing on ( (0, frac{pi}{2}) ).Since ( f(x) ) is increasing on ( (0, frac{pi}{2}) ), the larger the ( x ), the larger the value of ( f(x) ).Now, let's list the arguments in order:- ( frac{pi}{5} ) is approximately ( 0.628 ) radians.- ( 1 ) radian is approximately ( 1 ) radian.- ( frac{pi}{3} ) is approximately ( 1.047 ) radians.So, ordering them from smallest to largest:( frac{pi}{5} approx 0.628 < 1 < frac{pi}{3} approx 1.047 ).Since ( f(x) ) is increasing on ( (0, frac{pi}{2}) ), the function values will follow the same order:( fleft( frac{pi}{5} right) < f(1) < fleft( frac{pi}{3} right) ).But remember, ( fleft( frac{pi}{3} right) = fleft( -frac{pi}{3} right) ) because ( f(x) ) is even.So, substituting back:( fleft( frac{pi}{5} right) < f(1) < fleft( -frac{pi}{3} right) ).Therefore, the order from largest to smallest is:( fleft( -frac{pi}{3} right) > f(1) > fleft( frac{pi}{5} right) ).Looking back at the options:A: ( fleft( -frac{pi}{3} right) > f(1) > fleft( frac{pi}{5} right) )B: ( f(1) > fleft( -frac{pi}{3} right) > fleft( frac{pi}{5} right) )C: ( fleft( frac{pi}{5} right) > f(1) > fleft( -frac{pi}{3} right) )D: ( fleft( -frac{pi}{3} right) > fleft( frac{pi}{5} right) > f(1) )So, option A matches our conclusion.But wait, let me double-check my calculations to make sure I didn't make any mistakes.First, confirming that ( f(x) ) is even:( f(-x) = (-x) sin(-x) = (-x)(-sin x) = x sin x = f(x) ). Yep, that's correct.Next, the derivative ( f'(x) = sin x + x cos x ). Since both ( sin x ) and ( cos x ) are positive in ( (0, frac{pi}{2}) ), the derivative is positive, meaning the function is increasing in that interval. That seems right.Ordering of the arguments:( frac{pi}{5} approx 0.628 ), ( 1 approx 1 ), ( frac{pi}{3} approx 1.047 ). So, ( frac{pi}{5} < 1 < frac{pi}{3} ). Correct.Therefore, since the function is increasing, ( fleft( frac{pi}{5} right) < f(1) < fleft( frac{pi}{3} right) ). Which translates back to ( fleft( -frac{pi}{3} right) > f(1) > fleft( frac{pi}{5} right) ). So, option A is correct.I think that's solid. I don't see any errors in my reasoning.**Final Answer**boxed{A}

answer：First, I need to determine the total number of students in the high school. There are 400 freshmen, 320 sophomores, and 280 juniors.Adding these together:400 + 320 + 280 = 1000 students in total.Next, I know that the probability of each student being selected in the sample is 0.2. To find the sample size (n), I can use the formula:n = total number of students × probability of selection.Plugging in the numbers:n = 1000 × 0.2 = 200.Therefore, the required sample size is 200.

answer：First, convert the landing time in New York City to Chicago time. Since New York is one hour ahead of Chicago, 2:53 PM ET is equivalent to 1:53 PM CT.Next, calculate the duration from 9:17 AM to 1:53 PM in Chicago time:- From 9:17 AM to 12:00 PM is 2 hours and 43 minutes.- From 12:00 PM to 1:53 PM is 1 hour and 53 minutes.- Adding these together gives a total of 4 hours and 96 minutes. Since 96 minutes is 1 hour and 36 minutes, the total duration is 5 hours and 36 minutes.Finally, add the hours and minutes: ( h = 5 ) and ( m = 36 ), so ( h + m = 5 + 36 = 41 ).

answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that Sₙ is the sum of the first n terms of an arithmetic sequence {aₙ}. They give me that a₂ = 3 and a₆ = 11, and they're asking for S₇. The options are A: 13, B: 63, C: 49, D: 35.Alright, arithmetic sequences. I remember that in an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. So, the general formula for the nth term is aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.Given that a₂ = 3, so plugging into the formula, that's a₁ + (2-1)d = a₁ + d = 3. Similarly, a₆ = 11, which is a₁ + (6-1)d = a₁ + 5d = 11.So now I have two equations:1. a₁ + d = 32. a₁ + 5d = 11I can solve these two equations to find a₁ and d. Let's subtract the first equation from the second to eliminate a₁.(a₁ + 5d) - (a₁ + d) = 11 - 3a₁ + 5d - a₁ - d = 84d = 8d = 2Now that I have d, I can plug it back into the first equation to find a₁.a₁ + 2 = 3a₁ = 1So, the first term is 1 and the common difference is 2. Let me write out the first few terms to make sure I understand the sequence.a₁ = 1a₂ = 3a₃ = 5a₄ = 7a₅ = 9a₆ = 11a₇ = 13Okay, that makes sense. Each term increases by 2. Now, I need to find S₇, which is the sum of the first 7 terms.I remember the formula for the sum of the first n terms of an arithmetic sequence is Sₙ = n/2 * (a₁ + aₙ). So, for S₇, that would be S₇ = 7/2 * (a₁ + a₇).I already know a₁ is 1, and a₇ is 13, as I listed above. So plugging those in:S₇ = 7/2 * (1 + 13) = 7/2 * 14Hmm, 14 divided by 2 is 7, so 7 * 7 = 49.Wait, let me double-check that. Alternatively, I can use another formula for the sum: Sₙ = n/2 * [2a₁ + (n-1)d]. Let's try that with n=7, a₁=1, d=2.S₇ = 7/2 * [2*1 + (7-1)*2] = 7/2 * [2 + 12] = 7/2 * 14Again, 14 divided by 2 is 7, so 7*7=49. Okay, that's consistent.Looking back at the options, C is 49, so that must be the answer.But just to make sure I didn't make any mistakes, let me add up the terms manually:a₁ = 1a₂ = 3a₃ = 5a₄ = 7a₅ = 9a₆ = 11a₇ = 13Adding them up: 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25, 25 + 11 = 36, 36 + 13 = 49. Yep, that checks out.So, I'm confident that the answer is 49, which is option C.**Final Answer**boxed{C}

answer：Okay, so I have this problem about triangle DEF with sides 7, 10, and 13 units. I need to find the perimeter of a rectangle that has the same area as this triangle, and the rectangle has a length of 7 units. Hmm, let me break this down step by step.First, I should figure out the area of triangle DEF. Since the sides are 7, 10, and 13, it doesn't look like a right-angled triangle because 7² + 10² is 49 + 100, which is 149, and that's not equal to 13², which is 169. So, it's not a right triangle. That means I can't just use the simple base times height over two formula directly. I think I need to use Heron's formula here.Heron's formula says that the area of a triangle is the square root of [s(s - a)(s - b)(s - c)], where 's' is the semi-perimeter. Let me calculate the semi-perimeter first. The perimeter of the triangle is 7 + 10 + 13, which is 30 units. So, the semi-perimeter 's' is half of that, which is 15 units.Now, plugging the values into Heron's formula: Area = √[15(15 - 7)(15 - 10)(15 - 13)]. Let's compute each part inside the square root. 15 - 7 is 8, 15 - 10 is 5, and 15 - 13 is 2. So, it becomes √[15 * 8 * 5 * 2]. Multiplying those together: 15 * 8 is 120, 120 * 5 is 600, and 600 * 2 is 1200. So, the area is √1200.Wait, √1200 can be simplified. Let me see, 1200 is 400 * 3, and √400 is 20, so √1200 is 20√3. Okay, so the area of triangle DEF is 20√3 square units.Now, the rectangle has the same area, which is 20√3, and its length is 7 units. I need to find the width of the rectangle first. The area of a rectangle is length times width, so width = area / length. Plugging in the numbers, width = 20√3 / 7. That gives me the width.Now, to find the perimeter of the rectangle, I know the formula is 2*(length + width). The length is 7, and the width is 20√3 / 7. So, plugging in, perimeter = 2*(7 + 20√3 / 7). Let me compute that.First, 7 is the same as 49/7, so 49/7 + 20√3 /7 is (49 + 20√3)/7. Then, multiplying by 2 gives 2*(49 + 20√3)/7, which simplifies to (98 + 40√3)/7. Wait, 98 divided by 7 is 14, and 40√3 divided by 7 is (40/7)√3. So, the perimeter is 14 + (40√3)/7.Hmm, that seems a bit complicated, but I think that's correct. Let me double-check my steps. I found the semi-perimeter correctly, applied Heron's formula, simplified the square root correctly to 20√3. Then, for the rectangle, I used area equals length times width, solved for width, and then applied the perimeter formula. It all seems to add up. So, I think the perimeter is 14 + (40√3)/7 units.Wait, maybe I can write that differently. 40 divided by 7 is approximately 5.714, but since it's exact value, I should keep it as a fraction. So, 40√3 over 7 is fine. So, the perimeter is 14 plus 40√3 over 7. Alternatively, I can write it as (98 + 40√3)/7, but 14 is a whole number, so maybe it's better to leave it as 14 + (40√3)/7.Yeah, that seems right. I don't think I made any calculation errors. Heron's formula gave me the area, used that for the rectangle, found the width, then calculated the perimeter. Everything checks out.

answer：First, I recognize that the equation of a parabola in vertex form is ( y = a(x - h)^2 + k ), where ( (h, k) ) represents the vertex of the parabola. Given that the vertex is at ( (2, 5) ), I can substitute these values into the vertex form to get:[ y = a(x - 2)^2 + 5 ]Next, I know that the parabola passes through the point ( (3, 4) ). To find the value of ( a ), I substitute ( x = 3 ) and ( y = 4 ) into the equation:[ 4 = a(3 - 2)^2 + 5 ]Simplifying the equation:[ 4 = a(1)^2 + 5 ][ 4 = a + 5 ]Finally, I solve for ( a ) by subtracting 5 from both sides:[ a = 4 - 5 ][ a = -1 ]Therefore, the value of ( a ) is ( -1 ).