💡Okay, so I have this problem with two parts. Let me try to tackle them one by one. First, part (1): I need to find the polar coordinates of points A and B where curves C₁ and C₂ intersect. Curve C₁ has the polar equation ρ² cos 2θ = 8, and curve C₂ is given by θ = π/6. Alright, so since both curves intersect at points A and B, their polar coordinates must satisfy both equations. That means I can substitute θ = π/6 into the equation of C₁ to find ρ. So, substituting θ = π/6 into ρ² cos 2θ = 8, I get:ρ² cos(2 * π/6) = 8Simplify 2 * π/6 to π/3:ρ² cos(π/3) = 8I remember that cos(π/3) is 0.5, so:ρ² * 0.5 = 8Multiply both sides by 2:ρ² = 16Taking the square root of both sides gives ρ = ±4. So, the polar coordinates are (4, π/6) and (-4, π/6). But wait, in polar coordinates, negative radii can be represented by adding π to the angle. So, (-4, π/6) is the same as (4, π/6 + π) = (4, 7π/6). But the problem didn't specify whether to use positive radii or not, so maybe both points are valid as (4, π/6) and (-4, π/6). I think both are acceptable since they represent the same point but in opposite directions.Moving on to part (2): Curve C₁ intersects line l at points M and N. I need to find the length of segment MN.First, let me recall the equation of curve C₁. It's given in polar coordinates as ρ² cos 2θ = 8. I think it would be easier to work in Cartesian coordinates for this part because the line l is given in parametric form.I remember that in polar coordinates, ρ² = x² + y², and cos 2θ can be expressed in terms of x and y. Let me recall the identity:cos 2θ = (x² - y²) / (x² + y²)So, substituting into the equation of C₁:ρ² * (x² - y²) / (x² + y²) = 8But ρ² is x² + y², so:(x² + y²) * (x² - y²) / (x² + y²) = 8Simplify:x² - y² = 8So, the Cartesian equation of curve C₁ is x² - y² = 8. That's a hyperbola.Now, the parametric equations of line l are:x = 1 + (√3 / 2) ty = (1/2) tI need to find the points where this line intersects the hyperbola x² - y² = 8. So, I'll substitute x and y from the parametric equations into the hyperbola equation.Substituting x:x = 1 + (√3 / 2) tSo, x² = [1 + (√3 / 2) t]^2 = 1 + √3 t + (3/4) t²Substituting y:y = (1/2) tSo, y² = (1/4) t²Now, plug x² and y² into the hyperbola equation:[1 + √3 t + (3/4) t²] - [(1/4) t²] = 8Simplify:1 + √3 t + (3/4) t² - (1/4) t² = 8Combine like terms:1 + √3 t + (2/4) t² = 8Simplify (2/4) to (1/2):1 + √3 t + (1/2) t² = 8Subtract 8 from both sides:(1/2) t² + √3 t + 1 - 8 = 0Simplify 1 - 8 to -7:(1/2) t² + √3 t - 7 = 0Multiply both sides by 2 to eliminate the fraction:t² + 2√3 t - 14 = 0Now, I have a quadratic equation in terms of t: t² + 2√3 t - 14 = 0I can solve this using the quadratic formula. The quadratic formula is t = [-b ± √(b² - 4ac)] / (2a)Here, a = 1, b = 2√3, c = -14So, discriminant D = b² - 4ac = (2√3)² - 4*1*(-14) = 12 + 56 = 68So, t = [-2√3 ± √68] / 2Simplify √68: √68 = √(4*17) = 2√17So, t = [-2√3 ± 2√17] / 2Factor out 2 in numerator:t = [2(-√3 ± √17)] / 2 = -√3 ± √17So, the two solutions are t = -√3 + √17 and t = -√3 - √17These are the parameter values where the line intersects the hyperbola. So, points M and N correspond to these t values.Now, to find the length of segment MN, I can use the distance formula. Since both points lie on the line l, which is parametric, the distance between them can be found by the difference in their parameters multiplied by the magnitude of the direction vector.Wait, actually, the parametric equations are linear in t, so the distance between M and N is |t1 - t2| multiplied by the length of the direction vector.First, let's find |t1 - t2|:t1 = -√3 + √17t2 = -√3 - √17So, t1 - t2 = (-√3 + √17) - (-√3 - √17) = (-√3 + √17 + √3 + √17) = 2√17So, |t1 - t2| = 2√17Now, the direction vector of line l is given by the coefficients of t in the parametric equations:dx/dt = √3 / 2dy/dt = 1/2So, the direction vector is (√3 / 2, 1/2). The magnitude of this vector is:√[(√3 / 2)^2 + (1/2)^2] = √[(3/4) + (1/4)] = √[1] = 1So, the magnitude is 1. Therefore, the distance between M and N is |t1 - t2| * 1 = 2√17Alternatively, I could have used the distance formula directly by plugging t1 and t2 into the parametric equations and then computing the distance between the two points. But since the direction vector has magnitude 1, it's simpler to just take the difference in t.So, the length of segment MN is 2√17.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting from the parametric equations:x = 1 + (√3 / 2) ty = (1/2) tSubstituted into x² - y² = 8, expanded correctly, simplified to quadratic equation t² + 2√3 t - 14 = 0, solved using quadratic formula, got t = -√3 ± √17, then found |t1 - t2| = 2√17, and since the direction vector has magnitude 1, distance is 2√17. Seems correct.Alternatively, if I compute the distance using coordinates:For t1 = -√3 + √17:x1 = 1 + (√3 / 2)(-√3 + √17) = 1 + (-3/2 + (√51)/2) = 1 - 3/2 + (√51)/2 = (-1/2) + (√51)/2y1 = (1/2)(-√3 + √17) = (-√3)/2 + (√17)/2For t2 = -√3 - √17:x2 = 1 + (√3 / 2)(-√3 - √17) = 1 + (-3/2 - (√51)/2) = 1 - 3/2 - (√51)/2 = (-1/2) - (√51)/2y2 = (1/2)(-√3 - √17) = (-√3)/2 - (√17)/2Now, compute distance between (x1, y1) and (x2, y2):Δx = x1 - x2 = [(-1/2) + (√51)/2] - [(-1/2) - (√51)/2] = (√51)/2 + (√51)/2 = √51Δy = y1 - y2 = [(-√3)/2 + (√17)/2] - [(-√3)/2 - (√17)/2] = (√17)/2 + (√17)/2 = √17So, distance MN = √[(√51)^2 + (√17)^2] = √[51 + 17] = √68 = 2√17Yes, same result. So, that confirms the length is indeed 2√17.I think that's solid. So, part (1) gives us two points with polar coordinates (4, π/6) and (-4, π/6), and part (2) gives the length of MN as 2√17.

💡First, determine the distance Mercedes ran by doubling Jonathan's distance.Next, set up an equation for the total distance run by Mercedes and Davonte.Solve the equation to find Davonte's running distance.Finally, calculate the difference between Davonte's and Mercedes' distances to find how much farther Davonte ran.

💡Alright, so I have this problem here: If two factors of (2x^3 - hx + k) are (x + 2) and (x - 1), I need to find the value of (|2h - 3k|). The answer choices are given as (A) 4, (B) 3, (C) 2, (D) 1, and (E) 0.Hmm, okay. Let me try to figure this out step by step. I remember that if a polynomial has a factor like (x - c), then substituting (x = c) into the polynomial should give zero. This is called the Factor Theorem, which is a specific case of the Remainder Theorem. So, since (x + 2) is a factor, that means (x = -2) is a root, and similarly, (x - 1) is a factor, so (x = 1) is a root.Let me write down the polynomial: (2x^3 - hx + k). I need to plug in (x = -2) and (x = 1) into this polynomial and set them equal to zero because they are roots.First, let's substitute (x = -2):(2(-2)^3 - h(-2) + k = 0)Calculating each term:((-2)^3 = -8), so (2(-8) = -16).Then, (-h(-2) = 2h).And then we have (+k).So putting it all together:(-16 + 2h + k = 0)Let me write that as equation (1):(2h + k = 16) [Equation 1]Now, let's substitute (x = 1):(2(1)^3 - h(1) + k = 0)Calculating each term:(1^3 = 1), so (2(1) = 2).Then, (-h(1) = -h).And then (+k).So putting it all together:(2 - h + k = 0)Let me write that as equation (2):(-h + k = -2) [Equation 2]Now, I have two equations:1. (2h + k = 16)2. (-h + k = -2)I need to solve this system of equations to find the values of (h) and (k). Let me subtract equation (2) from equation (1) to eliminate (k):(2h + k - (-h + k) = 16 - (-2))Simplify:(2h + k + h - k = 16 + 2)Which simplifies to:(3h = 18)So, (h = 18 / 3 = 6).Now that I have (h = 6), I can substitute this back into one of the equations to find (k). Let's use equation (2):(-6 + k = -2)Adding 6 to both sides:(k = -2 + 6 = 4)So, (h = 6) and (k = 4).Now, the question asks for (|2h - 3k|). Let's compute that:First, compute (2h):(2 * 6 = 12)Then, compute (3k):(3 * 4 = 12)Now, subtract:(12 - 12 = 0)Taking the absolute value:(|0| = 0)So, the value of (|2h - 3k|) is 0.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with substituting (x = -2):(2(-2)^3 = 2*(-8) = -16)(-h*(-2) = 2h), and (+k). So, equation (1) is (-16 + 2h + k = 0), which simplifies to (2h + k = 16). That seems correct.Substituting (x = 1):(2(1)^3 = 2), (-h(1) = -h), and (+k). So, equation (2) is (2 - h + k = 0), which simplifies to (-h + k = -2). That also seems correct.Solving the system:Equation (1): (2h + k = 16)Equation (2): (-h + k = -2)Subtracting equation (2) from equation (1):(2h + k - (-h + k) = 16 - (-2))Which is (2h + k + h - k = 18), so (3h = 18), hence (h = 6). Then, plugging back into equation (2): (-6 + k = -2), so (k = 4). That seems correct.Calculating (|2h - 3k|):(2*6 = 12), (3*4 = 12), so (12 - 12 = 0), absolute value is 0. So, the answer is 0, which is option (E).I think that's correct. I don't see any mistakes in my calculations. So, I'm confident that the answer is 0.**Final Answer**boxed{E}

💡Okay, so I have this problem here: Jo reads at a steady pace. Her current book has 210 pages. Now, she is at page 90. An hour ago, she was at page 60. For how many hours will she be reading the book?Alright, let's see. I need to figure out how long Jo will be reading the book from now until she finishes it. She's currently on page 90, and an hour ago, she was on page 60. So, in one hour, she read from page 60 to page 90. That means she read 30 pages in one hour. So, her reading speed is 30 pages per hour.Now, the book has a total of 210 pages, and she's currently on page 90. So, she has 210 minus 90 pages left to read. Let me calculate that: 210 minus 90 is 120 pages remaining.Since she reads 30 pages each hour, to find out how many hours she'll need to finish the remaining 120 pages, I can divide the number of pages left by her reading speed. So, 120 pages divided by 30 pages per hour equals 4 hours.Wait, let me double-check that. If she reads 30 pages every hour, then in 4 hours, she would read 30 times 4, which is 120 pages. Adding that to her current page, 90 plus 120 equals 210, which is the total number of pages in the book. So, that makes sense.But hold on, the question is asking for how many hours she will be reading the book. Does that include the time she has already spent reading? She was at page 60 an hour ago, and now she's at page 90. So, she has already spent one hour reading. If she needs 4 more hours to finish, then the total time she will have spent reading the book is 1 hour plus 4 hours, which is 5 hours.Wait, but the way the question is phrased is a bit ambiguous. It says, "For how many hours will she be reading the book?" It could be interpreted as how much longer she will read from now until she finishes, which would be 4 hours. Alternatively, it could be asking for the total time she will spend reading the entire book, which would be 5 hours.Looking back at the problem, it says she is currently at page 90, and an hour ago, she was at page 60. So, she has already been reading for an hour. If the question is asking for the total time she will spend reading the book, including the time she has already spent, then the answer would be 5 hours. However, if it's asking for the remaining time she needs to finish the book from her current position, then it would be 4 hours.But in the initial solution, it calculated only the remaining time, which is 4 hours. Maybe the question is asking for the remaining time she needs to finish the book, not the total time. So, considering that, the answer is 4 hours.To be thorough, let's recast the problem. She reads at a steady pace, 30 pages per hour. She has 120 pages left. At 30 pages per hour, 120 divided by 30 is 4 hours. So, she needs 4 more hours to finish the book.Therefore, the final answer is 4 hours.

💡Alright, so I've got these two vector problems to solve. Let me take them one at a time. First, the problem says: Given vectors a = (2, -1) and b = (x, 1), where x is a real number. 1. Find the range of x for which the angle between a and b is acute.2. Find the value of x + y when 3a - 2b = (4, y).Okay, starting with the first part. I remember that the angle between two vectors can be determined using the dot product. If the dot product is positive, the angle is acute; if it's zero, the vectors are perpendicular; and if it's negative, the angle is obtuse. So, for the angle to be acute, the dot product of a and b must be greater than zero.Let me write that down. The dot product of a and b is:a · b = (2)(x) + (-1)(1) = 2x - 1.So, for the angle to be acute, 2x - 1 > 0. Solving for x:2x - 1 > 0 2x > 1 x > 1/2.So, the range of x is all real numbers greater than 1/2. That seems straightforward. I think that's the answer for the first part.Moving on to the second problem: Find the value of x + y when 3a - 2b = (4, y).Alright, let's compute 3a - 2b. First, I'll scale the vectors a and b by 3 and -2 respectively.3a = 3*(2, -1) = (6, -3). -2b = -2*(x, 1) = (-2x, -2).Now, adding these two results together:3a - 2b = (6, -3) + (-2x, -2) = (6 - 2x, -3 - 2) = (6 - 2x, -5).According to the problem, this equals (4, y). So, we can set up the equations:6 - 2x = 4 -5 = y.Let me solve the first equation for x:6 - 2x = 4 Subtract 6 from both sides: -2x = 4 - 6 -2x = -2 Divide both sides by -2: x = 1.And from the second component, we have y = -5.So, x + y = 1 + (-5) = -4.Wait, that seems straightforward too. Let me just double-check my calculations to make sure I didn't make any mistakes.For the first part: a · b = 2x - 1 > 0 leads to x > 1/2. That makes sense because if x is greater than 1/2, the dot product is positive, so the angle is acute.For the second part: 3a is (6, -3), -2b is (-2x, -2). Adding them gives (6 - 2x, -5). Setting this equal to (4, y) gives two equations: 6 - 2x = 4 and y = -5. Solving the first gives x = 1, and then x + y is 1 + (-5) = -4. That all checks out.I think I've got both parts right. The first part is just using the dot product condition for acute angles, and the second part is straightforward vector arithmetic. No complicated steps, just applying the definitions and solving the equations.Just to recap:1. For the angle to be acute, the dot product must be positive, so 2x - 1 > 0, which gives x > 1/2.2. Compute 3a - 2b, set it equal to (4, y), solve for x and y, then add them together to get x + y = -4.Yep, that all seems correct. I don't think I made any calculation errors, but let me just go through the steps one more time quickly.First problem:a · b = 2x - 1 > 0 2x > 1 x > 1/2.Second problem:3a = (6, -3) -2b = (-2x, -2) 3a - 2b = (6 - 2x, -5) Set equal to (4, y): 6 - 2x = 4 => x = 1 y = -5 x + y = 1 - 5 = -4.Yep, everything looks good. I think I'm confident with these answers.**Final Answer**1. The range of ( x ) is ( boxed{x > frac{1}{2}} ).2. The value of ( x + y ) is ( boxed{-4} ).

💡Okay, so I have this problem where I need to find two relatively prime positive integers, r and s, such that the fraction r/s equals this complicated expression: 2 times (sqrt(2) + sqrt(10)) divided by 5 times sqrt(3 + sqrt(5)). That is,[frac{r}{s} = frac{2(sqrt{2} + sqrt{10})}{5(sqrt{3 + sqrt{5}})}]Hmm, that looks pretty intimidating with all those square roots. I remember that sometimes when you have square roots in the denominator, you can rationalize them, but I'm not sure if that's the right approach here. Maybe I should try squaring both sides to get rid of some of the square roots? Let me think about that.If I square both sides, I can eliminate the square roots in the numerator and the denominator. So, let's try that. Squaring both sides gives:[left(frac{r}{s}right)^2 = left(frac{2(sqrt{2} + sqrt{10})}{5(sqrt{3 + sqrt{5}})}right)^2]Okay, so now I can square the numerator and the denominator separately. Let's start with the numerator:The numerator is 2 times (sqrt(2) + sqrt(10)). Squaring that gives:[(2(sqrt{2} + sqrt{10}))^2 = 4(sqrt{2} + sqrt{10})^2]Now, I need to expand (sqrt(2) + sqrt(10))^2. Using the formula (a + b)^2 = a^2 + 2ab + b^2, I get:[(sqrt{2})^2 + 2 cdot sqrt{2} cdot sqrt{10} + (sqrt{10})^2 = 2 + 2sqrt{20} + 10 = 12 + 4sqrt{5}]So, multiplying by 4, the squared numerator becomes:[4(12 + 4sqrt{5}) = 48 + 16sqrt{5}]Alright, that's the numerator squared. Now, let's work on the denominator. The denominator is 5 times sqrt(3 + sqrt(5)). Squaring that gives:[(5(sqrt{3 + sqrt{5}}))^2 = 25(sqrt{3 + sqrt{5}})^2]Simplifying (sqrt(3 + sqrt(5)))^2 is straightforward because squaring a square root cancels them out:[(sqrt{3 + sqrt{5}})^2 = 3 + sqrt{5}]So, the squared denominator becomes:[25(3 + sqrt{5}) = 75 + 25sqrt{5}]Now, putting the squared numerator and denominator together, we have:[left(frac{r}{s}right)^2 = frac{48 + 16sqrt{5}}{75 + 25sqrt{5}}]Hmm, this fraction still has square roots in both the numerator and the denominator. Maybe I can simplify this further by factoring out common terms. Let's see:Looking at the numerator, 48 + 16√5, I can factor out a 16:Wait, actually, 48 and 16 both have a common factor of 16? No, 48 divided by 16 is 3, and 16 divided by 16 is 1. So, factoring out 16 from the numerator gives:16(3 + √5)Similarly, in the denominator, 75 + 25√5, both terms are divisible by 25:25(3 + √5)So now, the fraction becomes:[frac{16(3 + sqrt{5})}{25(3 + sqrt{5})}]Oh, look at that! The (3 + √5) terms cancel out from the numerator and the denominator. So we're left with:[frac{16}{25}]Therefore, we have:[left(frac{r}{s}right)^2 = frac{16}{25}]Taking the square root of both sides, we get:[frac{r}{s} = frac{4}{5}]Since r and s are positive integers and relatively prime, the simplest form of the fraction 4/5 is already in its reduced form. So, r must be 4 and s must be 5.Wait, let me double-check my steps to make sure I didn't make any mistakes. I squared both sides correctly, expanded the squares properly, factored out the common terms, and then canceled out the (3 + √5) terms. Everything seems to check out. So, yeah, I think r is 4 and s is 5.**Final Answer**The relatively prime positive integers ( r ) and ( s ) are (boxed{4}) and (boxed{5}) respectively.**Final Answer**The relatively prime positive integers ( r ) and ( s ) are (boxed{4}) and (boxed{5}) respectively.