answer：Okay, so I have this triangle ABC, and the sides opposite angles A, B, and C are a, b, and c respectively. It's given that a < b < c, which probably means that angle A is the smallest and angle C is the largest because in a triangle, the larger side is opposite the larger angle. The problem also gives me this equation: (a)/(sin A) = (2b)/(√3). Hmm, that seems important. I remember from the Law of Sines that in any triangle, (a)/(sin A) = (b)/(sin B) = (c)/(sin C). So, all these ratios are equal. So, if (a)/(sin A) = (2b)/(√3), and also (a)/(sin A) = (b)/(sin B), then I can set those equal to each other. That would mean (b)/(sin B) = (2b)/(√3). Wait, if I do that, I can cancel out the b from both sides, assuming b isn't zero, which it can't be in a triangle. So, that leaves me with 1/(sin B) = 2/√3. Then, taking reciprocals, sin B = √3/2. Okay, sin B equals √3 over 2. I remember that sin 60° is √3/2, and sin 120° is also √3/2. But since the sides are a < b < c, that means angle A < angle B < angle C. So, angle B can't be 120° because that would make it larger than angle C, which contradicts the given side lengths. Therefore, angle B must be 60°, or π/3 radians. Alright, so that answers the first part. Angle B is 60 degrees. Now, moving on to the second part. They give me specific values: a = 2 and c = 3. I need to find the length of side b and the area of triangle ABC. Since I know angle B is 60°, maybe I can use the Law of Cosines to find b. The Law of Cosines says that b² = a² + c² - 2ac cos B. Let me plug in the values I have. So, b² = 2² + 3² - 2*2*3*cos 60°. Calculating each part: 2² is 4, 3² is 9, 2*2*3 is 12, and cos 60° is 0.5. So, putting it all together: b² = 4 + 9 - 12*0.5. Calculating that: 4 + 9 is 13, and 12*0.5 is 6. So, 13 - 6 is 7. Therefore, b² = 7, which means b is the square root of 7. So, b = √7. Now, for the area. I can use the formula for the area of a triangle: (1/2)*ab*sin C, but I need to know two sides and the included angle. Wait, I know sides a and c, and the included angle B. So, maybe I can use (1/2)*a*c*sin B. Let me check: a = 2, c = 3, and angle B is 60°, whose sine is √3/2. So, plugging in, the area is (1/2)*2*3*(√3/2). Let's compute that: (1/2)*2 is 1, 1*3 is 3, and 3*(√3/2) is (3√3)/2. So, the area is (3√3)/2. Wait, let me double-check my steps. For the Law of Cosines, I used the correct formula, plugged in the right values, and the calculations seem correct. For the area, I used the formula with two sides and the included angle, which is the right approach. The sine of 60° is indeed √3/2, so that part is correct too. I think that's it. So, side b is √7, and the area is (3√3)/2.**Final Answer**(1) The size of angle ( B ) is (boxed{dfrac{pi}{3}}).(2) The length of side ( b ) is (boxed{sqrt{7}}) and the area of ( triangle ABC ) is (boxed{dfrac{3sqrt{3}}{2}}).

answer：First, I need to determine the total age of the original class. Given that there are 18 students with an average age of 40 years, the total age is 18 multiplied by 40, which equals 720 years.Next, after 18 new students join, the total number of students becomes 36. The new average age is 36 years, so the total age of all 36 students is 36 multiplied by 36, resulting in 1296 years.To find the total age of the new students, I subtract the original total age from the new total age: 1296 years minus 720 years equals 576 years.Finally, to determine the average age of the 18 new students, I divide the total age of the new students by the number of new students: 576 years divided by 18 equals 32 years.

answer：Okay, so I have this math problem here: I need to express the sum of two fractions, each with numbers in different bases, and then convert the whole thing into base 10. The expression is (frac{254_8}{13_3} + frac{202_5}{22_4}). Hmm, that looks a bit complicated, but I think I can handle it step by step.First, I remember that when dealing with numbers in different bases, the best approach is to convert each number to base 10 before performing any operations. That way, everything is in the same system, and I can just add the results together. So, I'll start by converting each of these numbers individually.Let's break it down:1. **Convert (254_8) to base 10.** - The subscript 8 means it's an octal number. Each digit represents a power of 8. - Starting from the right, the digits are in the positions of (8^0), (8^1), (8^2), etc. - So, (254_8) can be expanded as: [ 2 times 8^2 + 5 times 8^1 + 4 times 8^0 ] - Calculating each term: - (2 times 64 = 128) - (5 times 8 = 40) - (4 times 1 = 4) - Adding them up: (128 + 40 + 4 = 172) - So, (254_8 = 172_{10})2. **Convert (13_3) to base 10.** - The subscript 3 means it's a ternary number. Each digit represents a power of 3. - Expanding (13_3): [ 1 times 3^1 + 3 times 3^0 ] - Wait a second, in base 3, the digits can only be 0, 1, or 2. Here, the rightmost digit is 3, which isn't valid in base 3. That must be a mistake. Maybe it's supposed to be (12_3) instead? Because 3 in base 3 would actually be represented as 10, similar to how 10 in base 10 is 10. - Let me check the original problem again. It says (13_3). Hmm, that's confusing. If it's indeed (13_3), then the digit 3 is invalid. Perhaps it's a typo, or maybe I need to adjust it. If I consider that maybe the number should be (12_3), then: [ 1 times 3^1 + 2 times 3^0 = 3 + 2 = 5 ] - Alternatively, if (13_3) is correct, then maybe it's a different base? But no, the subscript is 3, so it's base 3. I think there must be a mistake in the problem statement, or perhaps I need to interpret it differently. For now, I'll proceed assuming it's (12_3), which converts to 5 in base 10. I'll note this assumption.3. **Convert (202_5) to base 10.** - The subscript 5 means it's a base 5 number. - Expanding (202_5): [ 2 times 5^2 + 0 times 5^1 + 2 times 5^0 ] - Calculating each term: - (2 times 25 = 50) - (0 times 5 = 0) - (2 times 1 = 2) - Adding them up: (50 + 0 + 2 = 52) - So, (202_5 = 52_{10})4. **Convert (22_4) to base 10.** - The subscript 4 means it's a base 4 number. - Expanding (22_4): [ 2 times 4^1 + 2 times 4^0 ] - Calculating each term: - (2 times 4 = 8) - (2 times 1 = 2) - Adding them up: (8 + 2 = 10) - So, (22_4 = 10_{10})Now, with all the numbers converted to base 10, the original expression becomes:[frac{172}{5} + frac{52}{10}]Let me compute each fraction separately.- **Calculating (frac{172}{5}):** - Dividing 172 by 5: - 5 goes into 17 three times (3*5=15), remainder 2. - Bring down the 2: 22. - 5 goes into 22 four times (4*5=20), remainder 2. - Bring down the 0 (since we're dealing with decimals now): 20. - 5 goes into 20 four times exactly. - So, (frac{172}{5} = 34.4)- **Calculating (frac{52}{10}):** - Dividing 52 by 10 is straightforward. - (frac{52}{10} = 5.2)Now, adding the two results together:[34.4 + 5.2 = 39.6]So, the final result is 39.6 in base 10.Wait, but earlier I made an assumption about (13_3) being possibly (12_3). If that's not the case, and if (13_3) is indeed correct, then I need to figure out how to handle the digit 3 in base 3. In base 3, each digit must be less than 3, so having a digit 3 is invalid. Therefore, (13_3) is not a valid number in base 3. It should be (10_3) if it's supposed to represent 3 in base 10, or perhaps it's a typo. Since the problem states (13_3), I might need to reconsider.Alternatively, maybe the problem expects me to interpret (13_3) as a number where the digits are allowed to be higher, but that's not standard. In standard base notation, each digit must be between 0 and one less than the base. So, in base 3, digits can only be 0, 1, or 2. Therefore, (13_3) is incorrect. It could be that the number is actually (12_3), which is 5 in base 10, as I initially thought. Given that, I think my original calculation is correct, assuming (13_3) was a typo and should be (12_3). Therefore, the result of 39.6 stands.But just to be thorough, what if (13_3) is intended as a number with digits beyond the base? That's non-standard, but perhaps in some contexts, people allow digits equal to the base, interpreting them as carrying over. For example, (13_3) could be interpreted as (1 times 3^1 + 3 times 3^0), but since 3 is equal to the base, it would carry over. So, (3) in the units place would be equivalent to (1 times 3^1 + 0 times 3^0). Therefore, (13_3) would actually be (1 times 3^1 + 3 times 3^0 = 3 + 3 = 6), but since 3 is equal to the base, it's like having an extra 3, which would carry over to the next digit. So, (13_3) would actually be (20_3), which is (2 times 3^1 + 0 times 3^0 = 6 + 0 = 6). Wait, that's different from my initial assumption.So, if (13_3) is interpreted as (20_3), then it's 6 in base 10. That changes the calculation.Let me recalculate with this in mind.So, if (13_3 = 6_{10}), then:[frac{172}{6} + frac{52}{10}]Calculating each fraction:- (frac{172}{6}): - 6 goes into 17 two times (2*6=12), remainder 5. - Bring down the 2: 52. - 6 goes into 52 eight times (8*6=48), remainder 4. - Bring down the 0: 40. - 6 goes into 40 six times (6*6=36), remainder 4. - So, (frac{172}{6} = 28.666...) or (28.overline{6})- (frac{52}{10} = 5.2) as before.Adding them together:[28.overline{6} + 5.2 = 33.866...]Which is approximately 33.8667.But this contradicts my earlier result. So, now I'm confused because the interpretation of (13_3) affects the result significantly.Given that, perhaps the problem expects (13_3) to be treated as an invalid number, and perhaps it's a typo. Alternatively, maybe the problem expects me to recognize that (13_3) is invalid and adjust it accordingly.Alternatively, perhaps the problem is correct, and I need to interpret (13_3) as a number where the digits can exceed the base, which is non-standard, but perhaps in some contexts, it's allowed. In that case, (13_3) would be (1 times 3 + 3 = 6), as I calculated earlier.But in standard base notation, digits cannot exceed the base minus one. Therefore, (13_3) is invalid. So, perhaps the problem has a typo, and it's supposed to be (12_3), which is 5 in base 10.Given that, I think my initial calculation of 39.6 is correct, assuming (13_3) was a typo. However, if we take (13_3) as is, interpreting the digits beyond the base, we get a different result.But since in standard mathematics, digits in a base cannot exceed the base minus one, I think the problem might have an error. However, since the user provided the problem as is, I need to proceed with the assumption that (13_3) is correct, even if it's non-standard.Alternatively, perhaps the problem is correct, and I need to interpret (13_3) as a number where the digits can be equal to the base, which would mean carrying over. So, (13_3) would be equivalent to (20_3), which is 6 in base 10.Therefore, with that interpretation, the calculation becomes:[frac{172}{6} + frac{52}{10} = 28.overline{6} + 5.2 = 33.866...]Which is approximately 33.8667.But this is conflicting. I think the safest approach is to note that (13_3) is invalid and proceed with the assumption that it's a typo, and the intended number was (12_3), which is 5 in base 10, leading to the result of 39.6.Alternatively, if we proceed without assuming a typo, and treat (13_3) as is, even though it's non-standard, we get a different result.Given that, perhaps the problem expects me to recognize that (13_3) is invalid and adjust it accordingly. Therefore, I think the correct approach is to assume it's a typo and use (12_3) instead, leading to the result of 39.6.So, final answer: 39.6

answer：Okay, so I have this problem here: The equation of the tangent line to the graph of the function f(x) at x=2 is 2x + y - 3 = 0. I need to find the value of f(2) + f'(2). Hmm, let me think about how to approach this.First, I remember that the derivative of a function at a particular point gives the slope of the tangent line at that point. So, f'(2) should be equal to the slope of the tangent line given by the equation 2x + y - 3 = 0. Let me rewrite the equation of the tangent line in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Starting with 2x + y - 3 = 0, I can subtract 2x and add 3 to both sides to get y by itself.So, subtracting 2x from both sides: y = -2x + 3. Now, the equation is y = -2x + 3. From this, I can see that the slope (m) is -2. Therefore, f'(2) should be -2. That seems straightforward.Next, I need to find f(2). Since the tangent line touches the graph of f(x) at x=2, the point (2, f(2)) lies on both the function f(x) and the tangent line. So, if I plug x=2 into the equation of the tangent line, I should get the corresponding y-value, which is f(2).Let me do that. Plugging x=2 into y = -2x + 3:y = -2*(2) + 3 = -4 + 3 = -1.So, f(2) is -1. Now, the problem asks for f(2) + f'(2). I have f(2) = -1 and f'(2) = -2. Adding them together:-1 + (-2) = -3.Wait, that seems too simple. Let me double-check my steps.1. I converted the tangent line equation to slope-intercept form correctly: y = -2x + 3. So, the slope is indeed -2, meaning f'(2) = -2.2. Plugging x=2 into the tangent line equation: y = -2*(2) + 3 = -4 + 3 = -1. So, f(2) = -1.3. Adding f(2) and f'(2): -1 + (-2) = -3.Everything seems to check out. I don't think I made any mistakes here. The key was recognizing that the slope of the tangent line gives the derivative at that point and that the tangent line passes through the point (2, f(2)).I guess another way to think about it is that the tangent line just touches the graph of f(x) at x=2, so they share that point. Therefore, substituting x=2 into the tangent line equation gives me the y-coordinate, which is f(2). And the slope of the tangent line is the derivative at that point.So, yeah, I think I did it right. The answer should be -3.

answer：First, I need to solve the expression ( (125 + 17) times 8 ).I'll start by adding 125 and 17 together, which gives me 142.Next, I'll multiply 142 by 8 to get the final result.Now, I'll move on to the second expression: ( [458 - (85 + 28)] div 23 ).First, I'll add 85 and 28, which equals 113.Then, I'll subtract 113 from 458, resulting in 345.Finally, I'll divide 345 by 23 to obtain the final answer.

answer：First, I need to determine the equation of the line that passes through the points (2, 8) and (4, 14). To do this, I'll start by calculating the slope of the line.The slope (m) can be found using the formula:[m = frac{y_2 - y_1}{x_2 - x_1}]Plugging in the given points:[m = frac{14 - 8}{4 - 2} = frac{6}{2} = 3]So, the slope of the line is 3.Next, I'll use the point-slope form of the equation of a line to find the full equation. The point-slope form is:[y - y_1 = m(x - x_1)]Using the point (2, 8) and the slope m = 3:[y - 8 = 3(x - 2)]Simplifying this equation:[y - 8 = 3x - 6][y = 3x - 6 + 8][y = 3x + 2]So, the equation of the line is y = 3x + 2.To find where this line intersects the y-axis, I need to determine the value of y when x = 0. Plugging x = 0 into the equation:[y = 3(0) + 2 = 0 + 2 = 2]Therefore, the line intersects the y-axis at the point (0, 2).