💡First, I need to determine the total duration of the podcasts Roger has already planned.The first podcast is 45 minutes long.The second podcast is twice as long as the first, so it is 90 minutes.The fourth podcast is 1 hour, which is 60 minutes.The next podcast is also 1 hour, adding another 60 minutes.Adding these durations together: 45 minutes + 90 minutes + 60 minutes + 60 minutes = 255 minutes.Roger's total drive time is 6 hours, which is 360 minutes.To find the length of the third podcast, I subtract the total time of the other podcasts from the total drive time: 360 minutes - 255 minutes = 105 minutes.Therefore, the third podcast is 105 minutes long.

💡Okay, so I have this math problem here: I need to find the smallest positive integer x such that when I multiply it by 1512, the result is a power of a positive integer. They've told me that x is 49, but I need to figure out what the actual power is. Hmm, let's break this down step by step.First off, I think I need to understand what it means for a number to be a power of a positive integer. I guess that means the number should be something like n^k, where n and k are positive integers, right? So, 1512 times x should equal some number raised to an exponent.Alright, so to find x, I probably need to look at the prime factors of 1512. Prime factorization often helps with these kinds of problems. Let me try to factorize 1512.Starting with 2: 1512 divided by 2 is 756. Divided by 2 again is 378, and one more time by 2 is 189. So, that's three 2s. Now, 189 is divisible by 3: 189 divided by 3 is 63, divided by 3 again is 21, and one more time by 3 is 7. So, that's three 3s. Finally, 7 is a prime number, so we're done.So, the prime factors of 1512 are 2^3 * 3^3 * 7^1. Okay, got that.Now, I need to find the smallest x such that when I multiply it by 1512, the result is a perfect power. A perfect power means all the exponents in its prime factorization should be the same, right? So, I need to make sure that after multiplying by x, all the exponents are equal.Looking at the exponents in 1512: 2 has an exponent of 3, 3 also has an exponent of 3, but 7 only has an exponent of 1. To make all exponents equal, I need to increase the exponent of 7 to match the others. Since 2 and 3 are already at 3, I need to get 7 up to 3 as well.So, to make the exponent of 7 equal to 3, I need to multiply by 7^(3-1) = 7^2, which is 49. That makes sense because 49 is 7 squared, and multiplying 1512 by 49 will give me 2^3 * 3^3 * 7^3.Now, let's check what that number is. 2^3 is 8, 3^3 is 27, and 7^3 is 343. Multiplying these together: 8 * 27 is 216, and 216 * 343. Hmm, that's a big number. Let me calculate that.First, 216 * 300 is 64,800, and 216 * 43 is 9,312. Adding those together, 64,800 + 9,312 is 74,112. So, 1512 * 49 equals 74,112.Now, is 74,112 a perfect power? Well, since all the exponents in its prime factorization are 3, it should be a perfect cube. Let me verify that.If I take the cube root of 74,112, it should give me an integer. Let's see, 42^3 is 42 * 42 * 42. 42 * 42 is 1,764, and 1,764 * 42 is 74,088. Wait, that's not exactly 74,112. Did I do something wrong?Oh no, I think I made a mistake in my multiplication earlier. Let me recalculate 1512 * 49. 1512 * 49 is the same as 1512 * (50 - 1), which is 1512*50 - 1512*1. 1512*50 is 75,600, and subtracting 1512 gives me 75,600 - 1,512 = 74,088. Okay, so 1512 * 49 is actually 74,088, not 74,112. My earlier calculation was off.So, 74,088 is the correct product. Now, is 74,088 a perfect cube? Let's check. 42^3 is 74,088, as I calculated earlier. So, yes, 42 cubed is exactly 74,088. That means 1512 * 49 equals 42^3.Therefore, the power is 42 cubed, or 42^3.Wait, but just to make sure I didn't make any other mistakes, let me go through the steps again.1. Factorize 1512: 2^3 * 3^3 * 7^1.2. To make it a perfect power, all exponents should be equal.3. The exponents for 2 and 3 are already 3, so I need to increase the exponent of 7 to 3.4. Therefore, x should be 7^(3-1) = 7^2 = 49.5. Multiply 1512 by 49: 1512 * 49 = 74,088.6. Check if 74,088 is a perfect cube: 42^3 = 74,088.Everything checks out. So, x is indeed 49, and the power is 42 cubed.

💡Okay, so I have this problem with two curves, C₁ and C₂. I need to do two things: first, write the general equation for C₁ and the parametric equation for C₂. Second, find the maximum distance between points P on C₁ and Q on C₂. Hmm, let me start with part (1).For curve C₁, it's given in parametric form:x = √2 cosθy = 6 + √2 sinθI remember that parametric equations for circles usually have the form x = h + r cosθ and y = k + r sinθ, where (h, k) is the center and r is the radius. Comparing that to what I have here, it looks like the center is (0, 6) because there's no horizontal shift in x, and the y is shifted up by 6. The radius should be √2 since both x and y have √2 multiplied by the cosine and sine functions, respectively.So, to write the general equation, I can use the standard circle equation: (x - h)² + (y - k)² = r². Plugging in the values, that becomes x² + (y - 6)² = (√2)², which simplifies to x² + (y - 6)² = 2. Okay, that seems straightforward.Now, for curve C₂, it's given by the equation x²/10 + y² = 1. I recognize this as an ellipse. The standard form of an ellipse is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis. In this case, a² = 10 and b² = 1, so a = √10 and b = 1.To write the parametric equations for an ellipse, I know they are usually x = a cosφ and y = b sinφ, where φ is the parameter. So, substituting the values, the parametric equations for C₂ should be:x = √10 cosφy = sinφAlright, that takes care of part (1). Now, moving on to part (2), which is finding the maximum value of |PQ| where P is on C₁ and Q is on C₂.I need to maximize the distance between a point P on C₁ and a point Q on C₂. The distance formula between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²]. So, I need to express this distance in terms of the parameters θ and φ, and then find its maximum value.Let me denote point P as (√2 cosθ, 6 + √2 sinθ) and point Q as (√10 cosφ, sinφ). Then, the distance squared between P and Q is:(√10 cosφ - √2 cosθ)² + (sinφ - (6 + √2 sinθ))²Expanding this, it might get complicated, but maybe there's a smarter way. I remember that the maximum distance between two curves can sometimes be found by considering the distance between their centers and adding their radii if they are circles or ellipses.Wait, C₁ is a circle with center at (0, 6) and radius √2, and C₂ is an ellipse centered at the origin (0, 0) with semi-major axis √10 and semi-minor axis 1. So, maybe the maximum distance occurs when the points P and Q are aligned along the line connecting the centers of the two curves, extended outward.The distance between the centers is √[(0 - 0)² + (6 - 0)²] = 6. Then, if I go from the center of C₁, which is (0,6), in the direction away from the center of C₂, which is (0,0), and add the radius of C₁, which is √2, and the maximum extent of C₂ in that direction.But wait, C₂ is an ellipse. The maximum distance from the center of C₂ (which is at (0,0)) in the direction towards (0,6) would be along the y-axis. Since the semi-minor axis is 1, the farthest point on C₂ in the positive y-direction is (0,1). Similarly, the farthest point in the negative y-direction is (0,-1). But since we are going from (0,6) towards (0,0), the direction is along the negative y-axis. So, the farthest point on C₂ in that direction is (0,-1).Wait, but if we're trying to maximize the distance between P and Q, maybe we should consider the point on C₂ that is farthest from the center of C₁. The center of C₁ is at (0,6). The point on C₂ farthest from (0,6) would be the point on C₂ that is farthest in the positive y-direction, which is (0,1), but that's actually closer to (0,6). Wait, no, (0,1) is closer, but the point farthest in the negative y-direction is (0,-1), which is farther from (0,6).So, the distance from (0,6) to (0,-1) is 7. Then, adding the radius of C₁, which is √2, would give a total distance of 7 + √2. But wait, is that the maximum?Alternatively, maybe the maximum distance isn't just along the y-axis. Maybe there's a point on C₂ that, when connected to a point on C₁, gives a longer distance. Hmm.Let me think about this differently. The maximum distance between two points on two curves is the maximum of the distance function between any two points on the curves. So, perhaps I can parametrize both points and then find the maximum of the distance function.Let me denote P as (√2 cosθ, 6 + √2 sinθ) and Q as (√10 cosφ, sinφ). Then, the distance squared between P and Q is:(√10 cosφ - √2 cosθ)² + (sinφ - 6 - √2 sinθ)²To find the maximum distance, I can try to maximize this expression with respect to θ and φ.This seems a bit complicated because it's a function of two variables. Maybe I can fix one parameter and then maximize with respect to the other, but that might not be straightforward.Alternatively, perhaps I can consider the distance from a point on C₂ to the center of C₁, and then add the radius of C₁ to that distance. The maximum distance from a point on C₂ to the center of C₁ would be the maximum distance from (0,6) to any point on C₂, plus the radius of C₁.So, let's compute the maximum distance from (0,6) to any point on C₂. A point on C₂ is (√10 cosφ, sinφ). The distance squared from (0,6) to this point is:(√10 cosφ - 0)² + (sinφ - 6)² = 10 cos²φ + (sinφ - 6)²Expanding that, we get:10 cos²φ + sin²φ - 12 sinφ + 36Combine like terms:(10 cos²φ + sin²φ) - 12 sinφ + 36I know that cos²φ = 1 - sin²φ, so substituting:10(1 - sin²φ) + sin²φ - 12 sinφ + 36= 10 - 10 sin²φ + sin²φ - 12 sinφ + 36= 10 + 36 - 9 sin²φ - 12 sinφ= 46 - 9 sin²φ - 12 sinφSo, the distance squared is 46 - 9 sin²φ - 12 sinφ. To find the maximum distance, I need to maximize this expression with respect to φ.Let me denote f(φ) = 46 - 9 sin²φ - 12 sinφ. To find its maximum, I can treat it as a quadratic in sinφ. Let me set u = sinφ, so f(u) = 46 - 9u² - 12u.This is a quadratic function in u, which opens downward because the coefficient of u² is negative. The maximum occurs at the vertex. The vertex of a quadratic ax² + bx + c is at x = -b/(2a). So here, a = -9, b = -12.Thus, the maximum occurs at u = -(-12)/(2*(-9)) = 12/(-18) = -2/3.So, sinφ = -2/3. Plugging this back into f(u):f(-2/3) = 46 - 9*(-2/3)² - 12*(-2/3)= 46 - 9*(4/9) + 8= 46 - 4 + 8= 50So, the maximum distance squared is 50, which means the maximum distance is √50 = 5√2.Therefore, the maximum distance from a point on C₂ to the center of C₁ is 5√2. Then, adding the radius of C₁, which is √2, gives the maximum distance between P and Q as 5√2 + √2 = 6√2.Wait, but is this correct? Let me double-check. The maximum distance from (0,6) to C₂ is 5√2, and since C₁ has a radius of √2, the farthest point on C₁ from (0,6) would be in the same direction, so adding the radius gives the total maximum distance.Yes, that makes sense. So, the maximum |PQ| is 6√2.I think that's the answer. Let me just recap:1. Wrote the general equation for C₁ as x² + (y - 6)² = 2.2. Wrote the parametric equations for C₂ as x = √10 cosφ, y = sinφ.3. To find the maximum distance between P on C₁ and Q on C₂, considered the distance from (0,6) to C₂, found it's 5√2, then added the radius √2 to get 6√2.Yeah, that seems right. I don't see any mistakes in the calculations.

💡First, I need to calculate the price of the coat after applying the 30% discount. 30% of 120 is 36. Subtracting this from the original price gives me 84.Next, I'll apply the 10 coupon to the discounted price of 84, which brings the price down to 74.Then, I'll deduct the 5 rebate from 74, resulting in a price of 69.Finally, I'll add the 5% sales tax to 69. 5% of 69 is 3.45. Adding this tax gives me a total of 72.45.So, the total amount the shopper pays after all deductions and tax is 72.45.

💡First, I need to determine the total number of players on the cricket team. I know there are 37 throwers, and all of them are right-handed. The total number of right-handed players is 57, so there must be 57 minus 37, which is 20 right-handed players who are not throwers.Next, the remaining players on the team are divided into left-handed and right-handed, with one third being left-handed and two thirds being right-handed. Since there are 20 right-handed non-throwers, this represents two thirds of the non-throwers. To find the total number of non-throwers, I can set up the equation (2/3) * N = 20, where N is the number of non-throwers. Solving for N gives me N = 30.Finally, to find the total number of players on the team, I add the number of throwers (37) to the number of non-throwers (30), resulting in a total of 67 players.

💡Okay, so I have this problem where I need to find the range of values for the real number ( m ) such that the function ( f(x) = x^3 - 2mx^2 - mx ) is a "first-order ratio increasing function" but not a "second-order ratio increasing function." Hmm, let me break this down step by step.First, let me understand what a "first-order ratio increasing function" is. The problem says that if ( y = frac{f(x)}{x} ) is an increasing function on ( (0, +infty) ), then ( f(x) ) is called a first-order ratio increasing function. Similarly, for a second-order ratio increasing function, ( y = frac{f(x)}{x^2} ) needs to be increasing on the same interval.So, I need to analyze both ( frac{f(x)}{x} ) and ( frac{f(x)}{x^2} ) to determine the conditions on ( m ).Let me start with the first part: ( f(x) in Omega_1 ). That means ( g(x) = frac{f(x)}{x} ) must be increasing on ( (0, +infty) ).Calculating ( g(x) ):[g(x) = frac{f(x)}{x} = frac{x^3 - 2mx^2 - mx}{x} = x^2 - 2mx - m]So, ( g(x) = x^2 - 2mx - m ). To check if this is increasing, I need to compute its derivative and ensure it's non-negative for all ( x > 0 ).Calculating ( g'(x) ):[g'(x) = 2x - 2m]For ( g(x) ) to be increasing, ( g'(x) geq 0 ) for all ( x > 0 ). So:[2x - 2m geq 0 implies x geq m]But this inequality needs to hold for all ( x > 0 ). The smallest value ( x ) can take is approaching 0 from the right. So, as ( x ) approaches 0, the inequality becomes:[0 - 2m geq 0 implies -2m geq 0 implies m leq 0]Therefore, ( m ) must be less than or equal to 0 for ( g(x) ) to be increasing on ( (0, +infty) ).Alright, so that's the condition for ( f(x) in Omega_1 ). Now, I need to ensure that ( f(x) notin Omega_2 ). That means ( h(x) = frac{f(x)}{x^2} ) should not be an increasing function on ( (0, +infty) ).Let me compute ( h(x) ):[h(x) = frac{f(x)}{x^2} = frac{x^3 - 2mx^2 - mx}{x^2} = x - 2m - frac{m}{x}]So, ( h(x) = x - 2m - frac{m}{x} ). To determine if ( h(x) ) is increasing, I'll compute its derivative and see when it's non-negative.Calculating ( h'(x) ):[h'(x) = 1 - 0 + frac{m}{x^2} = 1 + frac{m}{x^2}]Wait, hold on. The derivative of ( x ) is 1, the derivative of ( -2m ) is 0, and the derivative of ( -frac{m}{x} ) is ( frac{m}{x^2} ). So, yes, ( h'(x) = 1 + frac{m}{x^2} ).For ( h(x) ) to be increasing, ( h'(x) geq 0 ) for all ( x > 0 ). Let's analyze this expression:[1 + frac{m}{x^2} geq 0]Since ( x^2 ) is always positive for ( x > 0 ), the sign of ( frac{m}{x^2} ) depends on ( m ). If ( m geq 0 ), then ( frac{m}{x^2} geq 0 ), so ( h'(x) = 1 + text{something non-negative} geq 1 ), which is always positive. Therefore, if ( m geq 0 ), ( h(x) ) is increasing.But we need ( h(x) ) to **not** be increasing. So, ( h'(x) ) must be negative for some ( x > 0 ). Let's see when that happens.If ( m < 0 ), then ( frac{m}{x^2} ) is negative. So, ( h'(x) = 1 + frac{m}{x^2} ). For ( h'(x) ) to be negative somewhere, we need:[1 + frac{m}{x^2} < 0 implies frac{m}{x^2} < -1 implies m < -x^2]But ( x ) is in ( (0, +infty) ), so ( x^2 ) can be any positive number. Therefore, for ( m < 0 ), there exists some ( x ) such that ( m < -x^2 ). Specifically, if we choose ( x ) such that ( x^2 > -m ), then ( m < -x^2 ) will hold, making ( h'(x) < 0 ).Therefore, if ( m < 0 ), ( h(x) ) is not increasing on ( (0, +infty) ). Putting it all together:- For ( f(x) in Omega_1 ), we need ( m leq 0 ).- For ( f(x) notin Omega_2 ), we need ( m < 0 ).The intersection of these two conditions is ( m < 0 ). So, the range of ( m ) is all real numbers less than 0.Wait, let me double-check. If ( m = 0 ), then ( f(x) = x^3 ). Then ( g(x) = x^2 ), which is increasing on ( (0, +infty) ), so ( f(x) in Omega_1 ). But ( h(x) = x ), which is also increasing. So, ( f(x) ) would be in ( Omega_2 ) as well, which we don't want. Hence, ( m = 0 ) is excluded. Therefore, ( m ) must be strictly less than 0.Yes, that makes sense. So, the range is ( (-infty, 0) ).**Final Answer**The range of values of ( m ) is boxed{(-infty, 0)}.