💡Okay, so I have these two vectors, a and b. Vector a is (2, 3) and vector b is (-2, 4). I need to find the cosine of the angle θ between them and then find the projection of vector b onto vector a. Hmm, let me think about how to approach this.First, for part (1), finding cos θ. I remember that the formula for the cosine of the angle between two vectors is the dot product of the vectors divided by the product of their magnitudes. So, cos θ equals (a · b) divided by (|a| |b|). That makes sense because the dot product relates to the angle between them, and the magnitudes are just the lengths of the vectors.Alright, so let me compute the dot product first. The dot product of a and b is (2)(-2) + (3)(4). Let me calculate that: 2 times -2 is -4, and 3 times 4 is 12. So, adding those together, -4 + 12 equals 8. Wait, that seems positive. So, the dot product is 8.Next, I need the magnitudes of vectors a and b. The magnitude of a vector is found by taking the square root of the sum of the squares of its components. So, for vector a, which is (2, 3), the magnitude |a| is sqrt(2² + 3²). That's sqrt(4 + 9), which is sqrt(13). Okay, so |a| is sqrt(13).For vector b, which is (-2, 4), the magnitude |b| is sqrt((-2)² + 4²). That's sqrt(4 + 16), which is sqrt(20). Hmm, sqrt(20) can be simplified because 20 is 4 times 5, so sqrt(4*5) is 2*sqrt(5). So, |b| is 2*sqrt(5).Now, putting it all together for cos θ: it's the dot product, which is 8, divided by the product of the magnitudes, which is sqrt(13) times 2*sqrt(5). So, that would be 8 divided by (2*sqrt(13)*sqrt(5)). Let me write that as 8/(2*sqrt(65)) because sqrt(13)*sqrt(5) is sqrt(65). Simplifying that, 8 divided by 2 is 4, so it's 4/sqrt(65).But usually, we rationalize the denominator, right? So, 4/sqrt(65) can be written as 4*sqrt(65)/65. So, cos θ is 4*sqrt(65)/65. That seems correct.Wait, let me double-check my calculations. The dot product was 2*(-2) + 3*4, which is -4 + 12, that's 8. The magnitudes: sqrt(4 + 9) is sqrt(13), and sqrt(4 + 16) is sqrt(20), which is 2*sqrt(5). So, 8 divided by (sqrt(13)*2*sqrt(5)) is indeed 8/(2*sqrt(65)) which simplifies to 4/sqrt(65), and then rationalizing gives 4*sqrt(65)/65. Okay, that seems solid.Moving on to part (2), finding the projection of vector b onto vector a. I remember that the projection of b onto a is given by the dot product of b and a divided by the magnitude of a squared, multiplied by vector a. But wait, the question just asks for the projection, not the vector projection. It says "the projection of b in the direction of a." So, that should be a scalar, right? So, it's |b| cos θ, which is the magnitude of b times the cosine of the angle between them.Wait, but earlier I found cos θ as 4*sqrt(65)/65, and |b| is 2*sqrt(5). So, multiplying those together, 2*sqrt(5) times 4*sqrt(65)/65. Let me compute that.First, multiply the constants: 2 times 4 is 8. Then, sqrt(5) times sqrt(65) is sqrt(5*65). 5 times 65 is 325, so sqrt(325). Hmm, sqrt(325) can be simplified because 325 is 25 times 13. So, sqrt(25*13) is 5*sqrt(13). So, sqrt(325) is 5*sqrt(13).Putting it all together, we have 8 times 5*sqrt(13) divided by 65. So, 8*5 is 40, so 40*sqrt(13)/65. Simplifying that fraction, 40 and 65 are both divisible by 5. 40 divided by 5 is 8, and 65 divided by 5 is 13. So, it becomes 8*sqrt(13)/13.Wait, but let me think again. Is the projection just |b| cos θ? Or is it (a · b)/|a|? Because sometimes the projection is defined as the scalar component, which is |b| cos θ, or as the vector projection, which is (a · b)/|a|² times a. But since the question says "the projection of b in the direction of a," I think it's referring to the scalar projection, which is |b| cos θ. So, that would be 2*sqrt(5) times 4*sqrt(65)/65, which is 8*sqrt(13)/13, as I calculated.Alternatively, if I compute it as (a · b)/|a|, that would be 8 divided by sqrt(13). But 8/sqrt(13) is equal to 8*sqrt(13)/13, which is the same as what I got before. So, either way, it's 8*sqrt(13)/13. So, that seems consistent.Wait, let me verify both methods. The scalar projection is |b| cos θ, which is 2*sqrt(5) times 4*sqrt(65)/65. Let's compute that:2*sqrt(5) * 4*sqrt(65) = 8*sqrt(325) = 8*5*sqrt(13) = 40*sqrt(13). Then, 40*sqrt(13)/65 simplifies to 8*sqrt(13)/13.Alternatively, the scalar projection is (a · b)/|a|, which is 8/sqrt(13) = 8*sqrt(13)/13. So, both methods give the same result, which is reassuring.Therefore, the projection of b in the direction of a is 8*sqrt(13)/13.Wait, but let me make sure I didn't make any calculation errors. So, starting again:For part (1):a · b = (2)(-2) + (3)(4) = -4 + 12 = 8.|a| = sqrt(2² + 3²) = sqrt(4 + 9) = sqrt(13).|b| = sqrt((-2)² + 4²) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5).So, cos θ = 8 / (sqrt(13) * 2*sqrt(5)) = 8 / (2*sqrt(65)) = 4 / sqrt(65) = 4*sqrt(65)/65.Yes, that's correct.For part (2):Projection of b onto a is |b| cos θ = 2*sqrt(5) * (4*sqrt(65)/65).Multiplying 2*sqrt(5) and 4*sqrt(65):2*4 = 8.sqrt(5)*sqrt(65) = sqrt(5*65) = sqrt(325) = 5*sqrt(13).So, 8*5*sqrt(13) = 40*sqrt(13).Then, 40*sqrt(13)/65 simplifies by dividing numerator and denominator by 5: 8*sqrt(13)/13.Alternatively, using the formula (a · b)/|a|:(a · b) is 8, |a| is sqrt(13), so 8/sqrt(13) = 8*sqrt(13)/13.Same result. So, that's correct.I think I've got it right. So, cos θ is 4*sqrt(65)/65, and the projection is 8*sqrt(13)/13.

💡First, I need to determine how many pounds of strawberries Rachel picked. The problem states that Rachel picked twice as much as Christine. Since Christine picked 10 pounds, Rachel picked:[10 , text{pounds} times 2 = 20 , text{pounds}]Next, I'll calculate the total amount of strawberries they picked together by adding Christine's and Rachel's amounts:[10 , text{pounds} + 20 , text{pounds} = 30 , text{pounds}]Finally, to find out how many pies they can make, I'll divide the total pounds of strawberries by the pounds needed per pie:[frac{30 , text{pounds}}{3 , text{pounds per pie}} = 10 , text{pies}]

💡To find the value of ( f(f(-1)) ) for the given piecewise function ( f(x) ), we proceed as follows:1. **Evaluate ( f(-1) ):** - Since ( -1 < 2 ), we use the second part of the function ( f(x) = 3 - x ). - Therefore, ( f(-1) = 3 - (-1) = 4 ).2. **Evaluate ( f(f(-1)) = f(4) ):** - Since ( 4 geq 2 ), we use the first part of the function ( f(x) = sqrt{x} ). - Therefore, ( f(4) = sqrt{4} = 2 ).Thus, the value of ( f(f(-1)) ) is 2.

💡First, I need to evaluate the expression (2359 + 180 div 60 times 3 - 359) by following the order of operations, which dictates that division and multiplication are performed before addition and subtraction.I start by calculating the division (180 div 60), which equals 3.Next, I multiply the result by 3, giving me (3 times 3 = 9).Now, I substitute this back into the original expression, resulting in (2359 + 9 - 359).Finally, I perform the addition and subtraction from left to right: (2359 + 9 = 2368), and then (2368 - 359 = 2009).Therefore, the final answer is 2009.

💡Okay, so I have this math problem here: If the value of the fraction 2 over (x minus 3) is 2, then what is the value of x? Hmm, let me try to figure this out step by step.Alright, the equation given is 2 divided by (x minus 3) equals 2. So, I can write that as:2 / (x - 3) = 2I need to solve for x. Hmm, how do I approach this? Well, I remember that when you have a fraction equal to a number, you can cross-multiply to get rid of the denominator. Let me try that.So, if I cross-multiply, I would multiply both sides of the equation by (x - 3). That should eliminate the denominator on the left side. Let's see:2 = 2 * (x - 3)Okay, so now I have 2 equals 2 times (x minus 3). That simplifies to:2 = 2x - 6Wait, is that right? Let me check. If I distribute the 2 on the right side, it should be 2 times x minus 2 times 3, which is 2x minus 6. Yeah, that seems correct.Now, I need to solve for x. So, I'll get all the constants on one side and the variable term on the other. Let me add 6 to both sides to get rid of the -6 on the right.2 + 6 = 2x - 6 + 6That simplifies to:8 = 2xOkay, so 8 equals 2x. Now, to find x, I just need to divide both sides by 2.8 / 2 = 2x / 2Which simplifies to:4 = xSo, x equals 4. Wait, let me double-check that to make sure I didn't make any mistakes. I'll substitute x = 4 back into the original equation to verify.Original equation:2 / (x - 3) = 2Substituting x = 4:2 / (4 - 3) = 2Simplify the denominator:2 / 1 = 2Which is true because 2 divided by 1 is indeed 2. So, that checks out.Hmm, did I consider all the steps correctly? Let me go through it again. Starting from the equation:2 / (x - 3) = 2Cross-multiplied to get:2 = 2(x - 3)Expanded to:2 = 2x - 6Added 6 to both sides:8 = 2xDivided by 2:4 = xYes, that all seems correct. I don't think I missed anything. So, x is definitely 4.Wait, but just to be thorough, are there any restrictions on x? Like, can x be 3? Because if x were 3, the denominator would be zero, which is undefined. But in this case, x is 4, which is fine because 4 minus 3 is 1, and that's a valid denominator. So, no issues there.Is there another way to approach this problem? Maybe by isolating x from the start? Let me try that.Starting with:2 / (x - 3) = 2I can rewrite this as:2 = 2(x - 3)Which is the same as before. So, whether I cross-multiply or rewrite it, I end up with the same equation. So, that reinforces that the steps are correct.Alternatively, I could subtract 2 from both sides, but that might complicate things. Let me see:2 / (x - 3) - 2 = 0But that seems more complicated because I'd have to combine the terms over a common denominator. Maybe it's better to stick with the cross-multiplication method.Another thought: what if I divide both sides by 2 first? Let's try that.Starting with:2 / (x - 3) = 2Divide both sides by 2:(2 / (x - 3)) / 2 = 2 / 2Simplify:(2 / (x - 3)) * (1/2) = 1Which simplifies to:1 / (x - 3) = 1Then, taking reciprocals on both sides:x - 3 = 1So, x = 4Oh, that's another way to do it, and it also gives x = 4. That's reassuring. So, whether I cross-multiply or divide both sides by 2 first, I still get the same result. That makes me more confident that x = 4 is the correct solution.I think I've covered all the bases here. I've solved the equation using two different methods, checked my answer by substituting it back into the original equation, and considered any potential restrictions on x. Everything seems to add up. So, I'm pretty sure that x equals 4 is the right answer.Just to recap:1. Start with the equation: 2 / (x - 3) = 22. Cross-multiply to eliminate the denominator: 2 = 2(x - 3)3. Distribute the 2 on the right side: 2 = 2x - 64. Add 6 to both sides: 8 = 2x5. Divide both sides by 2: x = 46. Verify by substituting x = 4 back into the original equation: 2 / (4 - 3) = 2, which is true.So, all steps lead to the conclusion that x is 4. I don't see any mistakes in my reasoning, and the answer makes sense in the context of the problem. Therefore, I'm confident that x equals 4 is the correct solution.

💡Okay, so I have this problem here asking which set of three line segments can form a triangle. The options are A, B, C, and D, each with different lengths. I remember that to form a triangle, the sum of any two sides must be greater than the third side. That’s called the triangle inequality theorem, right? Let me try to apply that to each option one by one.Starting with Option A: 1, 2, 3. Hmm, let's see. The two shorter sides are 1 and 2. If I add them together, 1 + 2 equals 3. Now, the longest side is also 3. So, 3 is not greater than 3; it's equal. I think for a triangle, the sum needs to be strictly greater, not just equal. So, I guess Option A can't form a triangle because 1 + 2 is not greater than 3.Moving on to Option B: 1, √2, 3. Okay, √2 is approximately 1.414. So, the two shorter sides are 1 and √2. Adding them together, 1 + 1.414 is about 2.414. The longest side is 3. Now, 2.414 is less than 3, so again, the sum of the two shorter sides isn't greater than the longest side. That means Option B also can't form a triangle.Next is Option C: 3, 4, 8. The two shorter sides are 3 and 4. Adding those together, 3 + 4 equals 7. The longest side is 8. 7 is less than 8, so once more, the sum of the two shorter sides isn't greater than the longest side. So, Option C can't form a triangle either.Finally, Option D: 4, 5, 6. Let's check this one. The two shorter sides are 4 and 5. Adding them together, 4 + 5 equals 9. The longest side is 6. Now, 9 is definitely greater than 6. So, in this case, the sum of the two shorter sides is greater than the longest side. That means Option D can form a triangle.Wait, just to make sure I didn't make a mistake, let me double-check each option quickly.Option A: 1 + 2 = 3, which isn't greater than 3. Correct, can't form a triangle.Option B: 1 + √2 ≈ 2.414, which is less than 3. Correct, can't form a triangle.Option C: 3 + 4 = 7, which is less than 8. Correct, can't form a triangle.Option D: 4 + 5 = 9, which is greater than 6. Correct, can form a triangle.Yeah, I think I did that right. So, only Option D works.