💡Okay, so I have this problem where a line intersects a circle at two points, A and B. The equation of the line is given as y equals the square root of 3 times x plus m, and the circle's equation is x squared plus (y minus 3) squared equals 6. The distance between points A and B is given as 2 times the square root of 2. I need to find the value of m.First, I remember that when a line intersects a circle, the points of intersection can be found by solving the system of equations. But since we know the distance between A and B, maybe there's a shortcut using the properties of circles and lines.I recall that the distance from the center of the circle to the line can be used to find the length of the chord AB. The formula for the length of a chord is 2 times the square root of (r squared minus d squared), where r is the radius of the circle and d is the distance from the center to the line.So, let me note down the given information:- Line: y = sqrt(3)x + m- Circle: x² + (y - 3)² = 6- Chord length AB: 2√2First, I need to find the radius of the circle. The equation is x² + (y - 3)² = 6, so the center is at (0, 3) and the radius r is sqrt(6).Next, I need to find the distance d from the center (0, 3) to the line y = sqrt(3)x + m. The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²).Let me rewrite the line equation in standard form. Starting with y = sqrt(3)x + m, subtract y from both sides to get sqrt(3)x - y + m = 0. So, A is sqrt(3), B is -1, and C is m.Now, plugging into the distance formula:d = |sqrt(3)*0 + (-1)*3 + m| / sqrt((sqrt(3))² + (-1)²)= |0 - 3 + m| / sqrt(3 + 1)= |m - 3| / 2So, d = |m - 3| / 2.Now, using the chord length formula: AB = 2√(r² - d²). We know AB is 2√2, so:2√2 = 2√(r² - d²)Divide both sides by 2:√2 = √(r² - d²)Square both sides:2 = r² - d²We know r² is 6, so:2 = 6 - d²Subtract 6 from both sides:-4 = -d²Multiply both sides by -1:4 = d²So, d² = 4, which means d = 2 or d = -2, but since distance can't be negative, d = 2.Wait, hold on. The distance d is |m - 3| / 2, so:|m - 3| / 2 = 2Multiply both sides by 2:|m - 3| = 4This gives two cases:1. m - 3 = 4 => m = 72. m - 3 = -4 => m = -1So, m can be 7 or -1.Looking back at the options:A: -7 or -1B: 1 or 7C: -1 or 7D: -7 or 1So, the correct answer is C: -1 or 7.Wait, but let me double-check my steps to make sure I didn't make a mistake.1. Calculated radius correctly as sqrt(6).2. Converted line equation to standard form correctly: sqrt(3)x - y + m = 0.3. Applied distance formula correctly: |sqrt(3)*0 - 1*3 + m| / sqrt(3 + 1) = |m - 3| / 2.4. Chord length formula: AB = 2√(r² - d²). Plugged in AB = 2√2, so √2 = √(6 - d²). Squared both sides: 2 = 6 - d². So, d² = 4, d = 2.5. Then, |m - 3| / 2 = 2 => |m - 3| = 4 => m = 7 or m = -1.Yes, that seems correct. So, the answer should be C: -1 or 7.**Final Answer**boxed{C}

💡Alright, so I need to find the sum of the prime factors of 77. Hmm, okay, let's start by understanding what prime factors are. Prime factors are the prime numbers that multiply together to give the original number. So, for 77, I need to break it down into prime numbers that multiply to 77.First, I know that 77 is an odd number, so it's not divisible by 2. Let me try dividing 77 by some small prime numbers to see if it works. Let's start with 3. 77 divided by 3 is approximately 25.67, which isn't a whole number, so 3 isn't a factor. Next, I'll try 5. 77 divided by 5 is 15.4, which also isn't a whole number. So, 5 isn't a factor either.Okay, moving on to the next prime number, which is 7. Let's see if 7 divides evenly into 77. 7 times 11 is 77, right? So, 7 times 11 equals 77. That means both 7 and 11 are prime factors of 77. Now, I should check if 11 is a prime number. Well, 11 is only divisible by 1 and itself, so yes, it's a prime number.So, the prime factors of 77 are 7 and 11. To find the sum of these prime factors, I just need to add them together. Let's do that: 7 plus 11 equals 18. Therefore, the sum of the prime factors of 77 is 18.Wait a minute, did I miss any other prime factors? Let me double-check. Starting from the smallest prime number, 2, I saw that 77 isn't divisible by 2. Then, 3 and 5 didn't work either. When I tried 7, it worked, giving me 11, which is also prime. So, I don't think there are any other prime factors besides 7 and 11. Therefore, adding them together gives me the correct sum of 18.I think I've covered all the necessary steps. I started by identifying the prime factors through division, confirmed that both factors are indeed prime, and then summed them up. It seems straightforward, and I don't see any mistakes in my calculations. So, I'm confident that the sum of the prime factors of 77 is 18.

💡Alright, so I have this problem here about converting "29.47 thousand" into scientific notation. The options are A, B, C, and D, each with different expressions. I need to figure out which one is correct.First, I remember that scientific notation is a way to express really big or really small numbers more manageable. It's usually written as a number between 1 and 10 multiplied by 10 raised to some power. So, for example, 1000 would be 1 x 10^3 in scientific notation.Now, the number given is "29.47 thousand." I need to convert that into a standard number first. "Thousand" means 1,000, so 29.47 thousand is the same as 29.47 x 1,000. Let me calculate that:29.47 x 1,000 = 29,470.Okay, so 29.47 thousand is 29,470. Now, I need to express 29,470 in scientific notation. To do that, I have to move the decimal point so that there's only one non-zero digit to the left of the decimal. In this case, 29,470 has the decimal at the end, so I need to move it four places to the left to get it after the 2.So, moving the decimal four places to the left:29,470 becomes 2.9470.But in scientific notation, we don't usually write trailing zeros after the decimal, so it's just 2.947.Now, since I moved the decimal four places to the left, that means I have to multiply by 10 raised to the power of 4 to keep the value the same. So, 2.947 x 10^4.Looking back at the options:A: 0.2947 x 10^6B: 2.947 x 10^4C: 2.947 x 10^5D: 29.47 x 10^4So, my calculation matches option B: 2.947 x 10^4.Wait, but I recall that sometimes people might interpret "thousand" differently. Let me double-check. "Thousand" is 10^3, so 29.47 thousand is 29.47 x 10^3, which is 29,470. Then, in scientific notation, it's 2.947 x 10^4. Yeah, that still seems right.Option C is 2.947 x 10^5, which would be 294,700, which is way too big. Option A is 0.2947 x 10^6, which is also 294,700. Option D is 29.47 x 10^4, which is 294,700 as well. So, none of those match 29,470 except for option B.I think I'm confident that option B is the correct answer.

💡First, I need to determine how much of the second alloy was used. I'll start by defining ( x ) as the amount of the second alloy in kilograms.Next, I'll calculate the amount of chromium in each alloy. The first alloy has 12% chromium, so the chromium from the first alloy is ( 0.12 times 15 ) kg. The second alloy has 10% chromium, so the chromium from the second alloy is ( 0.10 times x ) kg.The total weight of the new alloy is the sum of the weights of the two alloys, which is ( 15 + x ) kg. The new alloy has 10.6% chromium, so the total chromium in the new alloy is ( 0.106 times (15 + x) ) kg.I'll set up an equation to represent the total chromium in the new alloy:[0.12 times 15 + 0.10 times x = 0.106 times (15 + x)]Simplifying the equation:[1.8 + 0.10x = 1.59 + 0.106x]Subtracting ( 0.10x ) from both sides:[1.8 = 1.59 + 0.006x]Subtracting 1.59 from both sides:[0.21 = 0.006x]Finally, dividing both sides by 0.006:[x = frac{0.21}{0.006} = 35]So, 35 kilograms of the second alloy were used.

💡Alright, so I've got this problem about vectors and propositions. Let me try to figure it out step by step. First, the problem states that we have three non-zero vectors a, b, and c. There are two propositions given:- Proposition p: If a is parallel to b and b is parallel to c, then a is parallel to c.- Proposition q: If the dot product of a and b is zero, then a is perpendicular to b.Then, we're given four options (A to D) which are combinations of p and q using logical connectives. We need to determine which of these is false.Okay, let's break it down. I think the first thing I need to do is figure out whether propositions p and q are true or false on their own. Once I know that, I can evaluate the truth values of the combined propositions.Starting with proposition p: If a is parallel to b and b is parallel to c, then a is parallel to c. Hmm, parallelism in vectors means that one is a scalar multiple of the other. So, if a is parallel to b, there exists some scalar λ such that a = λb. Similarly, if b is parallel to c, there exists some scalar μ such that b = μc. So, substituting, a = λμc. Since λ and μ are scalars, their product is also a scalar, meaning a is a scalar multiple of c. Therefore, a is parallel to c. So, proposition p seems to be true.Now, proposition q: If the dot product of a and b is zero, then a is perpendicular to b. I remember that the dot product of two vectors is zero if and only if they are perpendicular (or orthogonal). So, if a · b = 0, then by definition, a and b are perpendicular. Therefore, proposition q is also true.Alright, so both p and q are true. Now, let's look at the options:A: p ∧ qB: ¬p ∨ ¬qC: ¬p ∨ qD: p ∨ ¬qWe need to find which one is false.Since both p and q are true, let's evaluate each option:A: p ∧ q. Since both p and q are true, their conjunction is true.B: ¬p ∨ ¬q. Since both p and q are true, their negations are false. So, false ∨ false is false.C: ¬p ∨ q. ¬p is false, and q is true. So, false ∨ true is true.D: p ∨ ¬q. p is true, and ¬q is false. So, true ∨ false is true.So, the only false proposition is B.Wait, let me double-check. If p is true, then ¬p is false. If q is true, then ¬q is false. So, in option B, we have false ∨ false, which is indeed false. The others all have at least one true component, making them true.Yeah, that seems right. So, the answer should be B.**Final Answer**boxed{B}

💡To solve the equation ( x + 2sqrt{x} - 8 = 0 ) using the method of substitution, follow these steps:1. **Introduce a substitution variable:** Let ( sqrt{x} = t ), where ( t geq 0 ). This implies that ( x = t^2 ).2. **Substitute into the original equation:** Replace ( x ) and ( sqrt{x} ) in the equation with ( t^2 ) and ( t ) respectively: [ t^2 + 2t - 8 = 0 ]3. **Solve the quadratic equation:** The equation ( t^2 + 2t - 8 = 0 ) is a quadratic in terms of ( t ). To solve it, factor the quadratic: [ (t - 2)(t + 4) = 0 ] This gives two potential solutions: [ t - 2 = 0 quad Rightarrow quad t = 2 ] [ t + 4 = 0 quad Rightarrow quad t = -4 ]4. **Discard invalid solutions:** Since ( t = sqrt{x} ) must be non-negative, ( t = -4 ) is not a valid solution. Thus, the only valid solution is ( t = 2 ).5. **Find ( x ) from ( t ):** Substitute ( t = 2 ) back into ( x = t^2 ): [ x = (2)^2 = 4 ]6. **Verify the solution:** Substitute ( x = 4 ) back into the original equation to ensure it satisfies the equation: [ 4 + 2sqrt{4} - 8 = 4 + 4 - 8 = 0 ] The equation holds true, confirming that ( x = 4 ) is the correct solution.**Final Answer:**The solution to the equation ( x + 2sqrt{x} - 8 = 0 ) is (boxed{4}).