💡Okay, so I've got this problem with two vectors, a and b. Vector a is (cos(3x/2), sin(3x/2)) and vector b is (cos(x/2), -sin(x/2)). The magnitude of their sum is given as 1, and I need to find x in the interval [0, π]. Hmm, let's see how to approach this.First, I remember that the magnitude of the sum of two vectors can be found using the formula |a + b|² = |a|² + |b|² + 2(a · b). Since |a + b| is given as 1, squaring both sides gives |a + b|² = 1² = 1. So, I can write:|a|² + |b|² + 2(a · b) = 1.Now, let's compute |a|² and |b|². Since both a and b are vectors with components involving sine and cosine, their magnitudes should be 1, right? Because cos²θ + sin²θ = 1. Let me verify that.For vector a: |a|² = cos²(3x/2) + sin²(3x/2) = 1. Similarly, for vector b: |b|² = cos²(x/2) + (-sin(x/2))² = cos²(x/2) + sin²(x/2) = 1. So, both |a| and |b| are 1. That simplifies things.So, plugging back into the equation:1 + 1 + 2(a · b) = 1.Simplifying that:2 + 2(a · b) = 1.Subtract 2 from both sides:2(a · b) = -1.Divide both sides by 2:a · b = -1/2.Alright, so the dot product of a and b is -1/2. Now, let's compute the dot product a · b. The dot product is given by:a · b = (cos(3x/2))(cos(x/2)) + (sin(3x/2))(-sin(x/2)).Simplify that:a · b = cos(3x/2)cos(x/2) - sin(3x/2)sin(x/2).Hmm, that expression looks familiar. It resembles the cosine of a sum formula. Remember, cos(A + B) = cos A cos B - sin A sin B. So, in this case, A is 3x/2 and B is x/2. Therefore:a · b = cos(3x/2 + x/2) = cos(2x).So, we have:cos(2x) = -1/2.Now, I need to solve for x in the interval [0, π]. Let's recall where cosine equals -1/2. The general solutions for cosθ = -1/2 are θ = 2π/3 + 2kπ and θ = 4π/3 + 2kπ, where k is any integer.But since 2x is the angle here, let's write:2x = 2π/3 + 2kπ or 2x = 4π/3 + 2kπ.Divide both sides by 2:x = π/3 + kπ or x = 2π/3 + kπ.Now, considering x must be in [0, π], let's find all possible solutions.For x = π/3 + kπ:- When k = 0: x = π/3, which is approximately 1.047 and is within [0, π].- When k = 1: x = π/3 + π = 4π/3, which is approximately 4.188, outside the interval.For x = 2π/3 + kπ:- When k = 0: x = 2π/3, which is approximately 2.094, still within [0, π].- When k = 1: x = 2π/3 + π = 5π/3, which is approximately 5.236, outside the interval.So, the solutions within [0, π] are x = π/3 and x = 2π/3.Wait, let me double-check if these solutions satisfy the original equation. Let's plug x = π/3 into |a + b|.Compute vector a: cos(3*(π/3)/2) = cos(π/2) = 0, sin(3*(π/3)/2) = sin(π/2) = 1. So, a = (0, 1).Compute vector b: cos((π/3)/2) = cos(π/6) = √3/2, -sin((π/3)/2) = -sin(π/6) = -1/2. So, b = (√3/2, -1/2).Add them: a + b = (0 + √3/2, 1 - 1/2) = (√3/2, 1/2).Compute the magnitude: √[(√3/2)² + (1/2)²] = √[(3/4) + (1/4)] = √[1] = 1. Perfect, that works.Now, x = 2π/3:Compute vector a: cos(3*(2π/3)/2) = cos(π) = -1, sin(3*(2π/3)/2) = sin(π) = 0. So, a = (-1, 0).Compute vector b: cos((2π/3)/2) = cos(π/3) = 1/2, -sin((2π/3)/2) = -sin(π/3) = -√3/2. So, b = (1/2, -√3/2).Add them: a + b = (-1 + 1/2, 0 - √3/2) = (-1/2, -√3/2).Compute the magnitude: √[(-1/2)² + (-√3/2)²] = √[(1/4) + (3/4)] = √[1] = 1. That also works.So, both x = π/3 and x = 2π/3 are valid solutions within the given interval. I think that's all. I don't see any mistakes in my calculations, so I'm confident these are the correct answers.**Final Answer**The solutions are boxed{dfrac{pi}{3}} and boxed{dfrac{2pi}{3}}.

💡Alright, so Laura has this charge account at the general store, and she's agreed to pay 9% compound annual interest. She charges 35 in January, and we need to figure out how much she'll owe six months later. Okay, let's break this down step by step.First off, compound interest. I remember that compound interest means that the interest is calculated on the initial principal and also on the accumulated interest from previous periods. So, it's not just simple interest where you earn the same amount each year; instead, it grows over time because you earn interest on your interest.The formula for compound interest is A = P(1 + r/n)^(nt), where:- A is the amount of money accumulated after n years, including interest.- P is the principal amount (the initial amount of money).- r is the annual interest rate (decimal).- n is the number of times that interest is compounded per year.- t is the time the money is invested or borrowed for, in years.In Laura's case:- P = 35 (the amount she charged in January).- r = 9% annual interest rate, which is 0.09 in decimal.- n = 12 because the interest is compounded monthly.- t = 0.5 years since we're calculating for six months.So, plugging these values into the formula:A = 35(1 + 0.09/12)^(12*0.5)First, let's calculate the monthly interest rate by dividing the annual rate by 12:0.09 / 12 = 0.0075Next, calculate the number of compounding periods:12 * 0.5 = 6Now, plug these back into the formula:A = 35(1 + 0.0075)^6Calculate the value inside the parentheses:1 + 0.0075 = 1.0075Now, raise this to the power of 6:1.0075^6 ≈ 1.045685Finally, multiply this by the principal amount:35 * 1.045685 ≈ 36.598975So, after six months, Laura will owe approximately 36.60.Wait a minute, let me double-check that. Sometimes when dealing with exponents, small errors can creep in. Let me recalculate 1.0075^6.Using a calculator:1.0075^1 = 1.00751.0075^2 = 1.0150561.0075^3 = 1.0226721.0075^4 = 1.0303751.0075^5 = 1.0381731.0075^6 = 1.046063Hmm, so it's approximately 1.046063, not 1.045685 as I previously thought. Let me adjust that.Now, multiply this by the principal amount:35 * 1.046063 ≈ 36.612205So, rounding to the nearest cent, that's 36.61.Wait, but earlier I got 36.60. Which one is correct? Let me check my calculations again.First, 0.09 divided by 12 is indeed 0.0075. Then, 12 times 0.5 is 6. So, raising 1.0075 to the 6th power:1.0075^6 = e^(6 * ln(1.0075)) ≈ e^(6 * 0.007457) ≈ e^(0.044742) ≈ 1.045685Wait, now I'm getting confused. Different methods are giving me slightly different results. Maybe it's due to rounding during intermediate steps.Let me try calculating 1.0075^6 step by step without rounding:1.0075^1 = 1.00751.0075^2 = 1.0075 * 1.0075 = 1.015056251.0075^3 = 1.01505625 * 1.0075 ≈ 1.022672661.0075^4 = 1.02267266 * 1.0075 ≈ 1.030375391.0075^5 = 1.03037539 * 1.0075 ≈ 1.038173081.0075^6 = 1.03817308 * 1.0075 ≈ 1.04606302So, it seems that 1.0075^6 is approximately 1.04606302.Therefore, multiplying by the principal:35 * 1.04606302 ≈ 36.6121957Rounding to the nearest cent, that's 36.61.But earlier, using the natural logarithm method, I got approximately 1.045685, leading to 36.60.This discrepancy is likely due to the precision of the intermediate steps. To resolve this, I'll use a more precise calculation.Using a calculator for higher precision:1.0075^6 ≈ 1.04606302Thus, 35 * 1.04606302 ≈ 36.6121957Rounding to the nearest cent, it's 36.61.But in financial calculations, sometimes they round to the nearest cent at each compounding period. Let's see if that affects the result.Calculating each month's interest:Month 1:Interest = 35 * 0.0075 = 0.2625Total = 35 + 0.2625 = 35.2625Month 2:Interest = 35.2625 * 0.0075 ≈ 0.26446875Total ≈ 35.2625 + 0.26446875 ≈ 35.52696875Month 3:Interest ≈ 35.52696875 * 0.0075 ≈ 0.2664522656Total ≈ 35.52696875 + 0.2664522656 ≈ 35.7934210156Month 4:Interest ≈ 35.7934210156 * 0.0075 ≈ 0.2684506576Total ≈ 35.7934210156 + 0.2684506576 ≈ 36.0618716732Month 5:Interest ≈ 36.0618716732 * 0.0075 ≈ 0.2704640375Total ≈ 36.0618716732 + 0.2704640375 ≈ 36.3323357107Month 6:Interest ≈ 36.3323357107 * 0.0075 ≈ 0.2724925178Total ≈ 36.3323357107 + 0.2724925178 ≈ 36.6048282285So, after six months, the total amount is approximately 36.6048, which rounds to 36.60.Wait, now I'm getting 36.60 when calculating month by month, but earlier with the formula, I got 36.61. This inconsistency is puzzling.Let me check the monthly calculations again:Month 1:35 * 1.0075 = 35.2625Month 2:35.2625 * 1.0075 = 35.52696875Month 3:35.52696875 * 1.0075 ≈ 35.7934210156Month 4:35.7934210156 * 1.0075 ≈ 36.0618716732Month 5:36.0618716732 * 1.0075 ≈ 36.3323357107Month 6:36.3323357107 * 1.0075 ≈ 36.6048282285Yes, that's correct. So, the total after six months is approximately 36.6048, which rounds to 36.60.But earlier, using the formula, I got 36.61. Why the difference?Ah, I see. When I used the formula, I calculated 1.0075^6 as approximately 1.046063, leading to 35 * 1.046063 ≈ 36.6121957, which rounds to 36.61.However, when calculating month by month, rounding at each step, the total is 36.60.This discrepancy arises because in the formula, we're using the exact value of 1.0075^6, whereas in the monthly calculation, we're rounding to the nearest cent each month, which slightly reduces the total amount.In financial calculations, it's common to round to the nearest cent at each compounding period. Therefore, the more accurate result, considering standard financial rounding practices, would be 36.60.But to ensure accuracy, let's use the formula with higher precision:1.0075^6 = e^(6 * ln(1.0075)) ≈ e^(6 * 0.00745705) ≈ e^(0.0447423) ≈ 1.045685Then, 35 * 1.045685 ≈ 36.598975, which rounds to 36.60.So, both methods agree when considering rounding at each step or using higher precision in the formula.Therefore, Laura will owe approximately 36.60 after six months.

💡Okay, so I have this problem where I need to compare two expressions: ( a^a b^b ) and ( (ab)^{frac{a+b}{2}} ). Both ( a ) and ( b ) are positive numbers, which is good because it means I don't have to worry about negative bases or anything like that. First, I want to understand what each expression represents. ( a^a ) is a term where the variable is both the base and the exponent, which is interesting. Similarly, ( b^b ) is the same but with ( b ). When I multiply them together, I get ( a^a b^b ). On the other hand, ( (ab)^{frac{a+b}{2}} ) is a single term raised to an exponent that's the average of ( a ) and ( b ). I think a good starting point is to take the ratio of these two expressions to see how they compare. So, let me compute ( frac{a^a b^b}{(ab)^{frac{a+b}{2}}} ). If this ratio is greater than 1, then ( a^a b^b ) is greater than ( (ab)^{frac{a+b}{2}} ), and if it's less than 1, then the opposite is true. If it's equal to 1, they are the same.Let me simplify this ratio step by step. First, ( (ab)^{frac{a+b}{2}} ) can be rewritten using the property of exponents that says ( (xy)^z = x^z y^z ). So, ( (ab)^{frac{a+b}{2}} = a^{frac{a+b}{2}} b^{frac{a+b}{2}} ).Now, substituting this back into the ratio, I have:[frac{a^a b^b}{a^{frac{a+b}{2}} b^{frac{a+b}{2}}}]I can separate the terms in the denominator:[frac{a^a}{a^{frac{a+b}{2}}} times frac{b^b}{b^{frac{a+b}{2}}}]Simplifying each fraction separately, using the property ( frac{x^m}{x^n} = x^{m-n} ):For the ( a ) terms:[a^{a - frac{a+b}{2}} = a^{frac{2a - a - b}{2}} = a^{frac{a - b}{2}}]Similarly, for the ( b ) terms:[b^{b - frac{a+b}{2}} = b^{frac{2b - a - b}{2}} = b^{frac{b - a}{2}} = b^{-frac{a - b}{2}} = left( frac{1}{b} right)^{frac{a - b}{2}}]Putting it all together, the ratio becomes:[a^{frac{a - b}{2}} times left( frac{1}{b} right)^{frac{a - b}{2}} = left( frac{a}{b} right)^{frac{a - b}{2}}]So, the ratio simplifies to ( left( frac{a}{b} right)^{frac{a - b}{2}} ). Now, I need to analyze this expression. Let's consider different cases based on the relationship between ( a ) and ( b ).**Case 1: ( a = b )**If ( a = b ), then ( frac{a}{b} = 1 ), and ( a - b = 0 ). So, the ratio becomes ( 1^0 = 1 ). This means ( a^a b^b = (ab)^{frac{a+b}{2}} ) when ( a = b ).**Case 2: ( a > b )**If ( a > b ), then ( frac{a}{b} > 1 ) and ( a - b > 0 ). Therefore, ( frac{a - b}{2} > 0 ). Since the base ( frac{a}{b} ) is greater than 1 and the exponent is positive, the entire expression ( left( frac{a}{b} right)^{frac{a - b}{2}} ) is greater than 1. This implies that ( a^a b^b > (ab)^{frac{a+b}{2}} ) when ( a > b ).**Case 3: ( a < b )**If ( a < b ), then ( frac{a}{b} < 1 ) and ( a - b < 0 ). Therefore, ( frac{a - b}{2} < 0 ). Now, the base ( frac{a}{b} ) is less than 1, and the exponent is negative. Remember that a number less than 1 raised to a negative exponent becomes greater than 1. For example, ( (1/2)^{-1} = 2 ). So, ( left( frac{a}{b} right)^{frac{a - b}{2}} = left( frac{b}{a} right)^{frac{b - a}{2}} ), which is greater than 1. Therefore, ( a^a b^b > (ab)^{frac{a+b}{2}} ) when ( a < b ).Putting all these cases together, we see that regardless of whether ( a ) is greater than, less than, or equal to ( b ), the ratio ( frac{a^a b^b}{(ab)^{frac{a+b}{2}}} ) is always greater than or equal to 1. Therefore, ( a^a b^b geq (ab)^{frac{a+b}{2}} ) for all positive ( a ) and ( b ).I should also consider if there are any special cases or if my reasoning might have missed something. For example, what if ( a ) or ( b ) is 1? Let's test ( a = 1 ) and ( b = 1 ). Then both expressions become ( 1^1 times 1^1 = 1 ) and ( (1 times 1)^{frac{1+1}{2}} = 1^1 = 1 ). So, they are equal, which fits with our earlier conclusion.Another test case: ( a = 2 ) and ( b = 1 ). Then ( a^a b^b = 2^2 times 1^1 = 4 times 1 = 4 ). On the other hand, ( (2 times 1)^{frac{2+1}{2}} = 2^{1.5} = sqrt{8} approx 2.828 ). So, 4 > 2.828, which again confirms that ( a^a b^b > (ab)^{frac{a+b}{2}} ).One more test case: ( a = 1 ) and ( b = 2 ). Then ( a^a b^b = 1^1 times 2^2 = 1 times 4 = 4 ). ( (1 times 2)^{frac{1+2}{2}} = 2^{1.5} approx 2.828 ). Again, 4 > 2.828, so the inequality holds.What if ( a ) and ( b ) are very large or very small? Let's try ( a = 10 ) and ( b = 1 ). Then ( a^a b^b = 10^{10} times 1^1 = 10,000,000,000 times 1 = 10,000,000,000 ). ( (10 times 1)^{frac{10+1}{2}} = 10^{5.5} = 10^{5} times 10^{0.5} approx 100,000 times 3.162 approx 316,200 ). Clearly, 10,000,000,000 > 316,200.Another test with ( a = 0.5 ) and ( b = 0.5 ). Then both expressions become ( (0.5)^{0.5} times (0.5)^{0.5} = (0.5)^{1} = 0.5 ). On the other hand, ( (0.5 times 0.5)^{frac{0.5 + 0.5}{2}} = (0.25)^{0.5} = 0.5 ). So, they are equal, which is consistent.What if ( a = 0.5 ) and ( b = 2 )? Then ( a^a b^b = (0.5)^{0.5} times 2^2 = frac{1}{sqrt{2}} times 4 approx 0.707 times 4 approx 2.828 ). ( (0.5 times 2)^{frac{0.5 + 2}{2}} = 1^{1.25} = 1 ). So, 2.828 > 1, which again holds.It seems like in all these test cases, the inequality holds true. Therefore, my reasoning seems solid.Another way to think about this is using logarithms. Taking the natural logarithm of both sides might help. Let me try that.Let ( L = ln(a^a b^b) ) and ( R = lnleft( (ab)^{frac{a+b}{2}} right) ).Compute ( L - R ):[L - R = ln(a^a b^b) - lnleft( (ab)^{frac{a+b}{2}} right) = ln(a^a) + ln(b^b) - left( frac{a + b}{2} ln(ab) right)]Simplify each term:[= a ln a + b ln b - frac{a + b}{2} (ln a + ln b)]Distribute the last term:[= a ln a + b ln b - frac{a}{2} ln a - frac{a}{2} ln b - frac{b}{2} ln a - frac{b}{2} ln b]Combine like terms:[= left( a - frac{a}{2} - frac{b}{2} right) ln a + left( b - frac{a}{2} - frac{b}{2} right) ln b]Simplify the coefficients:For the ( ln a ) term:[a - frac{a}{2} - frac{b}{2} = frac{a}{2} - frac{b}{2} = frac{a - b}{2}]For the ( ln b ) term:[b - frac{a}{2} - frac{b}{2} = frac{b}{2} - frac{a}{2} = frac{b - a}{2} = -frac{a - b}{2}]So, putting it all together:[L - R = frac{a - b}{2} ln a - frac{a - b}{2} ln b = frac{a - b}{2} (ln a - ln b) = frac{a - b}{2} ln left( frac{a}{b} right)]So, ( L - R = frac{a - b}{2} ln left( frac{a}{b} right) ).Now, let's analyze the sign of ( L - R ).If ( a = b ), then ( L - R = 0 ), so ( L = R ), meaning ( a^a b^b = (ab)^{frac{a+b}{2}} ).If ( a > b ), then ( a - b > 0 ) and ( frac{a}{b} > 1 ), so ( ln left( frac{a}{b} right) > 0 ). Therefore, ( L - R > 0 ), which implies ( L > R ), so ( a^a b^b > (ab)^{frac{a+b}{2}} ).If ( a < b ), then ( a - b < 0 ) and ( frac{a}{b} < 1 ), so ( ln left( frac{a}{b} right) < 0 ). Therefore, ( L - R = frac{a - b}{2} times text{negative} ). Since ( a - b ) is negative, multiplying by a negative gives a positive result. So, ( L - R > 0 ), which again implies ( a^a b^b > (ab)^{frac{a+b}{2}} ).This method also confirms the inequality holds in all cases.I think I've covered the problem from multiple angles: simplifying the ratio, testing specific cases, and using logarithms to analyze the difference. All methods lead to the same conclusion that ( a^a b^b ) is greater than or equal to ( (ab)^{frac{a+b}{2}} ) for all positive ( a ) and ( b ).Just to make sure, let me think about the behavior as ( a ) and ( b ) approach certain limits. For example, if ( a ) approaches 0 and ( b ) is fixed, what happens?Wait, ( a ) and ( b ) are both positive, so they can't be zero. But if ( a ) approaches 0, ( a^a ) approaches 1 because ( lim_{x to 0^+} x^x = 1 ). Similarly, ( b^b ) remains ( b^b ). So, ( a^a b^b ) approaches ( b^b ). On the other hand, ( (ab)^{frac{a + b}{2}} ) approaches ( (0 times b)^{frac{0 + b}{2}} = 0^{frac{b}{2}} = 0 ). So, ( a^a b^b ) approaches ( b^b ), which is greater than 0. Therefore, the inequality still holds.Similarly, if ( a ) approaches infinity and ( b ) is fixed, ( a^a ) grows extremely rapidly, much faster than ( (ab)^{frac{a + b}{2}} ). So, again, ( a^a b^b ) would be much larger.Therefore, even in these extreme cases, the inequality holds.I think I've thoroughly explored this problem. My conclusion is that ( a^a b^b ) is always greater than or equal to ( (ab)^{frac{a+b}{2}} ) for all positive ( a ) and ( b ), with equality only when ( a = b ).

💡First, I need to understand the custom operation ( x heartsuit y ), which is defined as ( frac{3x}{y} cdot frac{y}{x} ).Simplifying this operation:[x heartsuit y = frac{3x}{y} cdot frac{y}{x} = frac{3xy}{yx} = 3]This shows that ( x heartsuit y ) always equals 3, regardless of the values of ( x ) and ( y ), as long as neither ( x ) nor ( y ) is zero.Now, I'll apply this to the given expression ( (7 heartsuit (4 heartsuit 5)) heartsuit 2 ).First, evaluate the inner operation ( 4 heartsuit 5 ):[4 heartsuit 5 = 3]Substitute this result back into the expression:[(7 heartsuit 3) heartsuit 2]Next, evaluate ( 7 heartsuit 3 ):[7 heartsuit 3 = 3]Finally, substitute this result into the outer operation:[3 heartsuit 2 = 3]Therefore, the final result is:[boxed{3}]

💡Okay, so I have this complex number z equals 1 plus 2i, and I need to find its magnitude, which is denoted as |z|. Hmm, I remember that complex numbers have a real part and an imaginary part. In this case, the real part is 1 and the imaginary part is 2i. I think the magnitude of a complex number is like its distance from the origin on the complex plane. So, if I plot z = 1 + 2i on the complex plane, the real part 1 would be on the x-axis, and the imaginary part 2 would be on the y-axis. That means the point would be at (1, 2) in the coordinate system.Now, to find the distance from the origin to this point, I can use the Pythagorean theorem. The theorem says that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, the two sides would be the real part and the imaginary part of the complex number.So, let me write that down. The magnitude |z| should be the square root of (1 squared plus 2 squared). That is, |z| equals the square root of (1^2 + 2^2). Calculating that, 1 squared is 1, and 2 squared is 4. Adding them together gives 1 plus 4, which is 5. Therefore, |z| is the square root of 5.Wait, let me double-check that. If I have a complex number a + bi, then the magnitude is sqrt(a^2 + b^2). So, plugging in a = 1 and b = 2, I get sqrt(1 + 4) which is sqrt(5). Yeah, that seems right.I don't think I made any mistakes here. The real part is straightforward, and the imaginary part, even though it's multiplied by i, when squared, the i^2 becomes -1, but since we're dealing with magnitude, we're only concerned with the magnitude squared, which is positive. So, squaring 2i would give me 4, not -4, because we're considering the magnitude.So, adding 1 and 4 gives me 5, and the square root of 5 is approximately 2.236, but since the question just asks for |z|, I can leave it as sqrt(5). I think that's it. The magnitude of z = 1 + 2i is sqrt(5).

💡Alright, so I have this problem where points P and Q are on the graph of y = -1/2 x², and the triangle POQ is equilateral, with O being the origin. I need to find the length of one side of this triangle. Hmm, okay, let's break this down step by step.First, I know that an equilateral triangle has all sides equal. So, the distances from O to P, O to Q, and P to Q must all be the same. That gives me three equations to work with. Let me denote point P as (p, y_p) and point Q as (q, y_q). Since both points lie on the graph y = -1/2 x², I can express their y-coordinates in terms of their x-coordinates. So, y_p = -1/2 p² and y_q = -1/2 q².Now, the distance from O to P can be found using the distance formula: sqrt[(p - 0)² + (y_p - 0)²] which simplifies to sqrt[p² + y_p²]. Similarly, the distance from O to Q is sqrt[q² + y_q²]. Since PO = OQ, these two expressions must be equal. So, sqrt[p² + y_p²] = sqrt[q² + y_q²]. Squaring both sides to eliminate the square roots, I get p² + y_p² = q² + y_q².Substituting y_p and y_q with the expressions in terms of p and q, I have p² + (-1/2 p²)² = q² + (-1/2 q²)². Simplifying that, it becomes p² + (1/4 p⁴) = q² + (1/4 q⁴). So, p² + (1/4)p⁴ = q² + (1/4)q⁴.Hmm, okay, so that's one equation relating p and q. Now, I also know that the distance from P to Q must be equal to PO and OQ. So, let's write that distance as well. The distance PQ is sqrt[(q - p)² + (y_q - y_p)²]. Substituting y_q and y_p, it becomes sqrt[(q - p)² + (-1/2 q² - (-1/2 p²))²]. Simplifying further, that's sqrt[(q - p)² + ( -1/2 (q² - p²))²].Let me write that out: sqrt[(q - p)² + ( -1/2 (q² - p²))²]. Since we know that PQ must equal PO, which is sqrt[p² + (1/4)p⁴], we can set these equal to each other.So, sqrt[(q - p)² + ( -1/2 (q² - p²))²] = sqrt[p² + (1/4)p⁴]. Squaring both sides to eliminate the square roots, we get (q - p)² + ( -1/2 (q² - p²))² = p² + (1/4)p⁴.Expanding the left side: (q - p)² is q² - 2pq + p². The second term is ( -1/2 (q² - p²))² which is (1/4)(q² - p²)². So, expanding that, it's (1/4)(q⁴ - 2p² q² + p⁴).Putting it all together, the left side becomes q² - 2pq + p² + (1/4)(q⁴ - 2p² q² + p⁴). The right side is p² + (1/4)p⁴.So, setting them equal:q² - 2pq + p² + (1/4)(q⁴ - 2p² q² + p⁴) = p² + (1/4)p⁴.Let me simplify this equation. First, subtract p² from both sides:q² - 2pq + (1/4)(q⁴ - 2p² q² + p⁴) = (1/4)p⁴.Now, let's distribute the 1/4 on the left side:q² - 2pq + (1/4)q⁴ - (1/2)p² q² + (1/4)p⁴ = (1/4)p⁴.Subtract (1/4)p⁴ from both sides:q² - 2pq + (1/4)q⁴ - (1/2)p² q² = 0.Hmm, this is getting a bit complicated. Maybe I should consider symmetry here. If the triangle is equilateral with O at the origin, it's possible that points P and Q are symmetric with respect to the y-axis. That would mean that if P is (p, y_p), then Q would be (-p, y_p). Let me test this assumption.If Q is (-p, y_p), then q = -p and y_q = y_p. So, let's substitute q = -p into our earlier equations.First, let's go back to the equation p² + (1/4)p⁴ = q² + (1/4)q⁴. If q = -p, then q² = p² and q⁴ = p⁴. So, substituting, we get p² + (1/4)p⁴ = p² + (1/4)p⁴, which is always true. So, that condition is satisfied.Now, let's check the distance PQ. If Q is (-p, y_p), then the distance PQ is sqrt[(-p - p)² + (y_p - y_p)²] = sqrt[(-2p)² + 0] = sqrt[4p²] = 2|p|. But we also know that PQ must equal PO, which is sqrt[p² + (1/4)p⁴]. So, setting these equal:2|p| = sqrt[p² + (1/4)p⁴].Squaring both sides:4p² = p² + (1/4)p⁴.Subtract p² from both sides:3p² = (1/4)p⁴.Multiply both sides by 4:12p² = p⁴.Divide both sides by p² (assuming p ≠ 0, which it can't be because then P and Q would coincide with O, and the triangle wouldn't exist):12 = p².So, p² = 12, which means p = sqrt(12) or p = -sqrt(12). Simplifying sqrt(12), that's 2*sqrt(3). So, p = 2√3 or p = -2√3.Therefore, the coordinates of P and Q are (2√3, y_p) and (-2√3, y_p). Let's find y_p:y_p = -1/2 p² = -1/2 * (12) = -6.So, P is (2√3, -6) and Q is (-2√3, -6).Now, let's find the length of one side of the triangle. Let's compute PO:PO = sqrt[(2√3)^2 + (-6)^2] = sqrt[12 + 36] = sqrt[48] = 4√3.Similarly, PQ is the distance between (2√3, -6) and (-2√3, -6), which is sqrt[(-2√3 - 2√3)^2 + (-6 - (-6))^2] = sqrt[(-4√3)^2 + 0] = sqrt[48] = 4√3.And OQ is the same as PO, so it's also 4√3. So, all sides are equal, confirming that the triangle is equilateral.Therefore, the length of one side of triangle POQ is 4√3.**Final Answer**The length of one side of triangle (POQ) is boxed{4sqrt{3}}.