answer：First, I need to express the diameter of the Type A H1N1 influenza virus, which is 0.000000081 meters, in scientific notation.To do this, I'll move the decimal point to the right until there is only one non-zero digit to the left of the decimal point. Starting with 0.000000081, I move the decimal point 8 places to the right, resulting in 8.1.Since I moved the decimal point 8 places to the right, the exponent of 10 will be negative, specifically (10^{-8}).Combining these, the number in scientific notation is (8.1 times 10^{-8}) meters.Looking at the options provided, option B matches this expression.

answer：Okay, so I have this problem about a taxi driver operating east and west from the Drum Tower for two hours. The distances are given as +9, -3, -5, +4, -8, +6 kilometers. I need to figure out three things: the total distance traveled, the fuel cost, and the income excluding fuel costs. Let me try to break this down step by step.First, for part (1), the total distance traveled. Hmm, I think this means I need to add up all the distances, but since some are negative and some are positive, does that matter? Wait, no, because distance is a scalar quantity, so it's just the total amount of ground covered, regardless of direction. So I should take the absolute value of each segment and then add them all together. Let me write that out:Total distance = |+9| + |-3| + |-5| + |+4| + |-8| + |+6|Calculating each absolute value:|+9| = 9|-3| = 3|-5| = 5|+4| = 4|-8| = 8|+6| = 6Now, adding them up: 9 + 3 = 12; 12 + 5 = 17; 17 + 4 = 21; 21 + 8 = 29; 29 + 6 = 35.So the total distance traveled is 35 kilometers. That seems straightforward.Moving on to part (2), the fuel cost. The taxi consumes 0.08 liters per kilometer, and gasoline is 6 yuan per liter. So first, I need to find out how much fuel is consumed for the total distance, then multiply by the price per liter.Fuel consumed = total distance × consumption rateFuel consumed = 35 km × 0.08 L/km = 2.8 litersThen, fuel cost = fuel consumed × price per literFuel cost = 2.8 L × 6 yuan/L = 16.8 yuanOkay, so the fuel cost is 16.8 yuan.Now, part (3) is about the taxi's income excluding fuel costs. The fare structure is: 6 yuan for the first 3 kilometers, and 1.5 yuan per kilometer beyond that. I need to calculate the income from each segment and then sum them up, and then subtract the fuel cost to get the net income.Wait, but the segments are given as +9, -3, -5, +4, -8, +6. Do I need to consider each segment separately or the entire trip as a whole? The problem says "the recorded distances in order," so I think each segment is a separate trip. So each time the taxi moves, it's a new fare.So, for each segment, I need to calculate the fare:First segment: +9 kmFare = 6 yuan for the first 3 km, plus (9 - 3) × 1.5 yuan= 6 + 6 × 1.5 = 6 + 9 = 15 yuanSecond segment: -3 kmSince it's 3 km, which is exactly the first 3 km, so fare is 6 yuanThird segment: -5 kmAgain, first 3 km is 6 yuan, and the remaining 2 km is 2 × 1.5 = 3 yuanTotal fare: 6 + 3 = 9 yuanFourth segment: +4 kmFirst 3 km: 6 yuan, remaining 1 km: 1 × 1.5 = 1.5 yuanTotal fare: 6 + 1.5 = 7.5 yuanFifth segment: -8 kmFirst 3 km: 6 yuan, remaining 5 km: 5 × 1.5 = 7.5 yuanTotal fare: 6 + 7.5 = 13.5 yuanSixth segment: +6 kmFirst 3 km: 6 yuan, remaining 3 km: 3 × 1.5 = 4.5 yuanTotal fare: 6 + 4.5 = 10.5 yuanNow, adding up all the fares:15 + 6 + 9 + 7.5 + 13.5 + 10.5Let me compute step by step:15 + 6 = 2121 + 9 = 3030 + 7.5 = 37.537.5 + 13.5 = 5151 + 10.5 = 61.5So total income from fares is 61.5 yuan.Now, subtract the fuel cost to get the net income:Net income = total income - fuel cost = 61.5 - 16.8 = 44.7 yuanWait, but let me double-check the fare calculations for each segment to make sure I didn't make a mistake.First segment: +9 km6 + (9-3)*1.5 = 6 + 9 = 15. Correct.Second segment: -3 km6 yuan. Correct.Third segment: -5 km6 + (5-3)*1.5 = 6 + 3 = 9. Correct.Fourth segment: +4 km6 + (4-3)*1.5 = 6 + 1.5 = 7.5. Correct.Fifth segment: -8 km6 + (8-3)*1.5 = 6 + 7.5 = 13.5. Correct.Sixth segment: +6 km6 + (6-3)*1.5 = 6 + 4.5 = 10.5. Correct.Adding them up: 15 + 6 + 9 + 7.5 + 13.5 + 10.515 + 6 = 2121 + 9 = 3030 + 7.5 = 37.537.5 + 13.5 = 5151 + 10.5 = 61.5. Correct.Fuel cost: 16.8 yuan. So net income: 61.5 - 16.8 = 44.7 yuan.Yes, that seems right.So, summarizing:1) Total distance: 35 km2) Fuel cost: 16.8 yuan3) Net income: 44.7 yuanI think that's all. I hope I didn't miss anything.

answer：Okay, so I need to find the area of a parallelogram. I remember that the area of a parallelogram is calculated by multiplying the base by the height. But wait, let me make sure I understand what the base and height are. The base is one of the sides, and the height is the perpendicular distance from the base to the opposite side. In this problem, the base is given as 36 cm and the height is 24 cm. So, I think I just need to multiply these two numbers together to get the area. Let me write that down: Area = base × height. Plugging in the numbers, that would be Area = 36 cm × 24 cm. Hmm, 36 times 24... Let me do that multiplication step by step. 36 times 20 is 720, and 36 times 4 is 144. If I add those together, 720 plus 144 equals 864. So, the area should be 864 square centimeters. Wait, is there anything else I need to consider? Maybe the units? Yes, the base and height are both in centimeters, so the area will be in square centimeters. That makes sense. I don't think I need to worry about converting units here since they're already the same. Let me double-check my multiplication to be sure I didn't make a mistake. 36 times 24: breaking it down, 30 times 24 is 720, and 6 times 24 is 144. Adding those gives 864. Yep, that seems right. So, I think I've got it. The area of the parallelogram with a base of 36 cm and a height of 24 cm is 864 square centimeters.

answer：Okay, so I have this problem here: Given a real number ( a ), the derivative of the function ( f(x) = e^x + a cdot e^{-x} ) is an odd function. If the slope of a tangent line to the curve ( y = f(x) ) is ( frac{3}{2} ), I need to find the x-coordinate of the point where this tangent line touches the curve.Hmm, let me break this down. First, I know that an odd function satisfies the condition ( f(-x) = -f(x) ) for all ( x ) in its domain. So, if the derivative ( f'(x) ) is odd, then ( f'(-x) = -f'(x) ). That should help me find the value of ( a ).Let me start by finding the derivative of ( f(x) ). The function is ( f(x) = e^x + a cdot e^{-x} ). The derivative of ( e^x ) is ( e^x ), and the derivative of ( a cdot e^{-x} ) is ( -a cdot e^{-x} ) because of the chain rule. So, putting that together, the derivative ( f'(x) ) is:( f'(x) = e^x - a cdot e^{-x} )Now, since ( f'(x) ) is an odd function, it must satisfy ( f'(-x) = -f'(x) ). Let me compute ( f'(-x) ):( f'(-x) = e^{-x} - a cdot e^{x} )According to the property of odd functions, this should equal ( -f'(x) ). Let's compute ( -f'(x) ):( -f'(x) = -e^x + a cdot e^{-x} )So, setting ( f'(-x) = -f'(x) ):( e^{-x} - a cdot e^{x} = -e^x + a cdot e^{-x} )Hmm, let me rearrange terms to see if I can solve for ( a ). Let's bring all terms to one side:( e^{-x} - a cdot e^{x} + e^x - a cdot e^{-x} = 0 )Factor terms with ( e^x ) and ( e^{-x} ):( (1 - a) e^{-x} + (-a + 1) e^x = 0 )Wait, that simplifies to:( (1 - a)(e^{-x} + e^x) = 0 )Since ( e^{-x} + e^x ) is never zero for any real ( x ), the only way this equation holds is if ( 1 - a = 0 ). Therefore, ( a = 1 ).Alright, so now I know that ( a = 1 ). Therefore, the function becomes:( f(x) = e^x + e^{-x} )And its derivative is:( f'(x) = e^x - e^{-x} )Now, the problem states that the slope of the tangent line is ( frac{3}{2} ). The slope of the tangent line at a point is given by the derivative at that point. So, I need to solve for ( x ) in the equation:( f'(x) = frac{3}{2} )Substituting the expression for ( f'(x) ):( e^x - e^{-x} = frac{3}{2} )Hmm, this looks like a hyperbolic sine function, because ( sinh(x) = frac{e^x - e^{-x}}{2} ). So, maybe I can rewrite the equation in terms of ( sinh(x) ):( 2 sinh(x) = frac{3}{2} )Which simplifies to:( sinh(x) = frac{3}{4} )But I'm not sure if that helps me directly. Maybe I should solve the equation ( e^x - e^{-x} = frac{3}{2} ) directly.Let me set ( y = e^x ). Then, ( e^{-x} = frac{1}{y} ). Substituting into the equation:( y - frac{1}{y} = frac{3}{2} )Multiply both sides by ( y ) to eliminate the denominator:( y^2 - 1 = frac{3}{2} y )Bring all terms to one side:( y^2 - frac{3}{2} y - 1 = 0 )Multiply both sides by 2 to eliminate the fraction:( 2y^2 - 3y - 2 = 0 )Now, this is a quadratic equation in terms of ( y ). Let me use the quadratic formula to solve for ( y ):( y = frac{3 pm sqrt{9 + 16}}{4} = frac{3 pm sqrt{25}}{4} = frac{3 pm 5}{4} )So, the solutions are:1. ( y = frac{3 + 5}{4} = frac{8}{4} = 2 )2. ( y = frac{3 - 5}{4} = frac{-2}{4} = -frac{1}{2} )But ( y = e^x ) must be positive because the exponential function is always positive. So, ( y = -frac{1}{2} ) is not a valid solution. Therefore, ( y = 2 ).Since ( y = e^x ), we have:( e^x = 2 )Taking the natural logarithm of both sides:( x = ln(2) )So, the x-coordinate of the point of tangency is ( ln(2) ).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the derivative, I found ( a = 1 ) correctly because the derivative had to be odd. Then, setting the derivative equal to ( frac{3}{2} ) led me to solve the equation ( e^x - e^{-x} = frac{3}{2} ). I substituted ( y = e^x ), transformed the equation into a quadratic, solved it, and found ( y = 2 ). Taking the natural log gives ( x = ln(2) ). That seems correct.Just to be thorough, let me plug ( x = ln(2) ) back into the derivative to verify:( f'( ln(2) ) = e^{ln(2)} - e^{-ln(2)} = 2 - frac{1}{2} = frac{4}{2} - frac{1}{2} = frac{3}{2} )Yes, that checks out. So, the slope at ( x = ln(2) ) is indeed ( frac{3}{2} ). Therefore, my answer is correct.**Final Answer**The x-coordinate of the point of tangency is boxed{ln 2}.

answer：First, I need to find the common difference ( d ) of the arithmetic sequence. I know that the fifth term ( a_5 = 10 ) and the sum of the first three terms ( a_1 + a_2 + a_3 = 3 ).Since the sequence is arithmetic, the terms can be expressed as:[a_1 = a_1][a_2 = a_1 + d][a_3 = a_1 + 2d][a_5 = a_1 + 4d = 10]From the sum of the first three terms:[a_1 + (a_1 + d) + (a_1 + 2d) = 3][3a_1 + 3d = 3][a_1 + d = 1][a_1 = 1 - d]Substituting ( a_1 = 1 - d ) into the equation for ( a_5 ):[(1 - d) + 4d = 10][1 + 3d = 10][3d = 9][d = 3]Now, using the common difference ( d = 3 ), I can find ( a_8 ):[a_8 = a_1 + 7d][a_8 = (1 - 3) + 7 times 3][a_8 = -2 + 21][a_8 = 19]

answer：Okay, so I have this problem here where I need to find the general equation of a curve given its parametric equations. The parametric equations are:x = sin α - cos αy = sin 2αAnd α is the parameter. The options given are:A: y = x² + 1B: y = -x² + 1C: y = -x² + 1, with x in [-√2, √2]D: y = x² + 1, with x in [-√2, √2]Alright, so I need to eliminate the parameter α and find a relationship between x and y. Let me think about how to approach this.First, I remember that when dealing with parametric equations, one common method is to express both x and y in terms of the parameter and then try to eliminate the parameter. In this case, both x and y are given in terms of α, so maybe I can find a way to relate them.Looking at the first equation, x = sin α - cos α. I wonder if I can square both sides to make use of the Pythagorean identity. Let me try that.So, if I square both sides of x = sin α - cos α, I get:x² = (sin α - cos α)²Expanding the right side, it becomes:x² = sin² α - 2 sin α cos α + cos² αHmm, I know that sin² α + cos² α = 1, so I can substitute that in:x² = 1 - 2 sin α cos αNow, looking at the second equation, y = sin 2α. I remember that sin 2α is equal to 2 sin α cos α. So, y = 2 sin α cos α.Wait, that means that 2 sin α cos α is equal to y. So, in the equation for x², I have:x² = 1 - yBecause 2 sin α cos α is y, so substituting that in:x² = 1 - ySo, if I rearrange this equation to solve for y, I get:y = 1 - x²Which can also be written as:y = -x² + 1Alright, so that gives me the equation y = -x² + 1. Now, looking at the options, both B and C have this equation. So, I need to figure out if there are any restrictions on x.The problem mentions that α is the parameter, but it doesn't specify any restrictions on α. However, I should consider the range of possible values for x. Since x = sin α - cos α, let's find the maximum and minimum values of x.I know that sin α and cos α both have ranges between -1 and 1. So, sin α - cos α can range between -√2 and √2. Let me verify that.The expression sin α - cos α can be rewritten using the amplitude-phase form. Specifically, it can be expressed as √2 sin(α - 45°) or √2 cos(α + 45°). Both forms have an amplitude of √2, so the maximum value is √2 and the minimum is -√2.Therefore, x must lie in the interval [-√2, √2].So, putting it all together, the equation is y = -x² + 1, with x in [-√2, √2]. That matches option C.Wait a second, let me double-check my steps to make sure I didn't make any mistakes.1. I started by squaring x = sin α - cos α, which gave me x² = sin² α - 2 sin α cos α + cos² α.2. Then, I used the identity sin² α + cos² α = 1, so x² = 1 - 2 sin α cos α.3. Recognized that 2 sin α cos α is sin 2α, which is equal to y. So, x² = 1 - y.4. Rearranged to get y = 1 - x², which is the same as y = -x² + 1.5. Then, considered the range of x, which is between -√2 and √2 because x = sin α - cos α can be rewritten as √2 sin(α - 45°), whose amplitude is √2.Everything seems to check out. So, the general equation is y = -x² + 1, with x in [-√2, √2], which is option C.I think that's the correct answer. I don't see any errors in my reasoning, so I'm confident with this choice.**Final Answer**boxed{C}