answer：Alright, so I've got this trigonometry problem here: "Given that sin α + cos α equals 1/2, find the value of sin 2α." The options are A: -3/8, B: -3/4, C: sqrt(7)/4, and D: -sqrt(7)/4. Hmm, okay, let's see how to approach this.First off, I remember that there are some trigonometric identities that relate sin and cos of an angle to sin of double angles. Specifically, the double-angle identity for sine: sin 2α = 2 sin α cos α. So, if I can find the value of sin α cos α, I can just multiply it by 2 to get sin 2α.But how do I find sin α cos α from the given equation sin α + cos α = 1/2? Maybe I can square both sides of the equation. Let me try that.So, if I square the left side, I get (sin α + cos α)^2. Expanding that, it should be sin²α + 2 sin α cos α + cos²α. And squaring the right side, (1/2)^2 is 1/4. So, putting it all together:sin²α + 2 sin α cos α + cos²α = 1/4.Now, I remember another identity: sin²α + cos²α = 1. So, I can substitute that into the equation:1 + 2 sin α cos α = 1/4.Subtracting 1 from both sides gives:2 sin α cos α = 1/4 - 1.Calculating the right side, 1/4 minus 1 is the same as 1/4 minus 4/4, which equals -3/4. So now I have:2 sin α cos α = -3/4.But wait, sin 2α is equal to 2 sin α cos α, right? So that means:sin 2α = -3/4.Looking back at the options, B is -3/4, so that must be the answer. But just to make sure I didn't make any mistakes, let me double-check my steps.1. I started with sin α + cos α = 1/2.2. Squared both sides to get (sin α + cos α)^2 = (1/2)^2.3. Expanded the left side to sin²α + 2 sin α cos α + cos²α.4. Replaced sin²α + cos²α with 1.5. Subtracted 1 from both sides to isolate 2 sin α cos α.6. Found that 2 sin α cos α equals -3/4, which is sin 2α.Everything seems to check out. I don't see any errors in my calculations or reasoning. So, yeah, I'm confident that the answer is B: -3/4.**Final Answer**boxed{B}

answer：First, I recognize that Ella has two dogs, and both dogs consume the same amount of dog food each day.The total amount of dog food consumed by both dogs together is 0.25 scoops per day.To find out how much each dog eats individually, I need to divide the total amount by the number of dogs.So, I divide 0.25 scoops by 2 dogs, which equals 0.125 scoops per dog per day.Therefore, each dog eats 0.125 scoops of dog food daily.

answer：First, I need to calculate the sums of the numerators and denominators in both fractions.For the first fraction, the numerator is (3 + 6 + 9 = 18) and the denominator is (2 + 4 + 8 = 14). This gives me the fraction (frac{18}{14}).For the second fraction, the numerator is (2 + 4 + 8 = 14) and the denominator is (3 + 6 + 9 = 18). This gives me the fraction (frac{14}{18}).Next, I simplify both fractions to their lowest terms. The first fraction (frac{18}{14}) simplifies to (frac{9}{7}) by dividing both the numerator and the denominator by 2.The second fraction (frac{14}{18}) simplifies to (frac{7}{9}) by dividing both the numerator and the denominator by 2.Now, I need to subtract the second fraction from the first fraction: (frac{9}{7} - frac{7}{9}).To perform this subtraction, I find a common denominator for the two fractions. The least common denominator of 7 and 9 is 63.I convert (frac{9}{7}) to (frac{81}{63}) by multiplying both the numerator and the denominator by 9.I convert (frac{7}{9}) to (frac{49}{63}) by multiplying both the numerator and the denominator by 7.Finally, I subtract the two fractions: (frac{81}{63} - frac{49}{63} = frac{32}{63}).Therefore, the value of the expression is (frac{32}{63}).

answer：First, I need to determine the incorrect total sum of the observations by multiplying the mean by the number of observations.Next, I'll calculate the difference between the correct observation and the incorrect observation.Then, I'll adjust the incorrect total sum by adding this difference to obtain the corrected total sum.Finally, I'll find the corrected mean by dividing the corrected total sum by the number of observations.

answer：First, I need to determine how many books the boys received in total. Since the total number of books is 375 and the girls received 225 books, I subtract the girls' books from the total to find the boys' share.Next, I divide the number of books the boys received by the number of boys to find out how many books each boy got.Knowing that each boy received 15 books, and assuming the distribution is equal for all students, each girl also received 15 books. To find the number of girls, I divide the total number of books the girls received by the number of books each girl got.This calculation gives me the total number of girls in the class.

answer：Alright, so I've got this math problem here, and I need to figure out the relationship between two statements: "x < 1" and "x^2 - 4x + 3 > 0". The options are about whether one is a sufficient condition, necessary condition, both, or neither for the other. Hmm, okay, let's break this down step by step.First, I need to understand what each statement means. The first one is straightforward: "x < 1" means that the value of x is less than 1. The second statement is a quadratic inequality: "x^2 - 4x + 3 > 0". I think I need to solve this inequality to see what values of x satisfy it.So, let's solve "x^2 - 4x + 3 > 0". To do that, I'll first try to factor the quadratic expression. Let me see, I need two numbers that multiply to 3 and add up to -4. Hmm, -1 and -3? Yeah, that works because (-1) * (-3) = 3 and (-1) + (-3) = -4. So, the quadratic factors to (x - 1)(x - 3) > 0.Now, to solve the inequality (x - 1)(x - 3) > 0, I need to find the intervals where this product is positive. The roots of the equation (x - 1)(x - 3) = 0 are x = 1 and x = 3. These roots divide the number line into three intervals:1. x < 12. 1 < x < 33. x > 3I'll test each interval to see where the product is positive.For the first interval, x < 1, let's pick x = 0. Plugging in, we get (0 - 1)(0 - 3) = (-1)(-3) = 3, which is positive. So, this interval satisfies the inequality.For the second interval, 1 < x < 3, let's choose x = 2. Plugging in, we get (2 - 1)(2 - 3) = (1)(-1) = -1, which is negative. So, this interval does not satisfy the inequality.For the third interval, x > 3, let's take x = 4. Plugging in, we get (4 - 1)(4 - 3) = (3)(1) = 3, which is positive. So, this interval also satisfies the inequality.Putting it all together, the solution to the inequality is x < 1 or x > 3.Now, let's compare this to the first statement, "x < 1". It seems that "x < 1" is part of the solution set for the inequality, but the inequality also includes values where x > 3. So, "x < 1" is sufficient to make the inequality true because if x is less than 1, then the inequality holds. However, it's not necessary because the inequality can also hold when x is greater than 3.In other words, "x < 1" guarantees that "x^2 - 4x + 3 > 0" is true, but there are other values of x (specifically, x > 3) that also make the inequality true without satisfying "x < 1".So, putting it all together, "x < 1" is a sufficient condition for "x^2 - 4x + 3 > 0", but it's not a necessary condition because there are other values of x that satisfy the inequality without being less than 1.I think that makes sense. Let me just double-check to make sure I didn't make any mistakes in my factoring or interval testing. Factoring the quadratic: yes, (x - 1)(x - 3) is correct. Testing intervals: x = 0 gives positive, x = 2 gives negative, and x = 4 gives positive. Yep, that all checks out.So, my conclusion is that "x < 1" is a sufficient but not necessary condition for "x^2 - 4x + 3 > 0".