💡Okay, so I have this problem about a right circular cone that's 8 inches tall with a base radius of 5 inches. The surface of the cone, excluding the base, is painted. Then, a plane parallel to the base cuts the cone into two parts: a smaller cone C and a frustum F. The height of the smaller cone C is 3 inches. I need to find the ratio k of the painted surface areas of C to F, express it as m/n where m and n are coprime, and then find m + n.Alright, let's start by understanding the problem. The original cone has a height of 8 inches and a base radius of 5 inches. The surface area that's painted is just the lateral (or curved) surface area, not including the base. So, when the cone is cut by a plane parallel to the base, we get a smaller cone C on top and a frustum F at the bottom.Since the cut is parallel to the base, the smaller cone C is similar to the original cone. The height of C is 3 inches, so the ratio of similarity is 3/8. That means all the linear dimensions of C are 3/8 of the original cone. So, the radius of the base of C should be (3/8)*5 = 15/8 inches. That makes sense.Now, the painted surface area of a cone is its lateral surface area, which is πrl, where r is the radius and l is the slant height. So, I need to find the slant heights of both the original cone and the smaller cone C.For the original cone, the slant height l can be found using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle with legs equal to the height (8 inches) and the radius (5 inches). So, l = sqrt(8^2 + 5^2) = sqrt(64 + 25) = sqrt(89). That's approximately 9.433 inches, but I'll keep it as sqrt(89) for exactness.Similarly, for the smaller cone C, the slant height l_c can be found using the same method. The height is 3 inches, and the radius is 15/8 inches. So, l_c = sqrt(3^2 + (15/8)^2) = sqrt(9 + 225/64). Let me compute that:9 is 576/64, so 576/64 + 225/64 = 801/64. Therefore, l_c = sqrt(801/64) = sqrt(801)/8. Hmm, 801 factors into 9*89, so sqrt(801) is 3*sqrt(89). So, l_c = (3*sqrt(89))/8 inches.Okay, so the lateral surface area of the original cone is π*5*sqrt(89). But wait, actually, the painted surface area is only the lateral surface, so the original painted area is π*5*sqrt(89). But since we're cutting it into C and F, we need to find the painted areas of both C and F.For the smaller cone C, its lateral surface area is π*(15/8)*l_c. Plugging in l_c, that's π*(15/8)*(3*sqrt(89)/8) = π*(45*sqrt(89))/64.Now, for the frustum F, the painted surface area is a bit trickier. A frustum is like a cone with the top cut off, so its lateral surface area is the difference between the lateral surface areas of the original cone and the smaller cone C. Alternatively, it can be calculated using the formula π*(R + r)*s, where R and r are the radii of the two bases, and s is the slant height of the frustum.Wait, let me confirm that formula. Yes, the lateral surface area of a frustum is π*(R + r)*s, where s is the slant height of the frustum. So, I need to find the slant height of the frustum F.The slant height of the frustum can be found by subtracting the slant height of the smaller cone from the slant height of the original cone. But actually, since the frustum is part of the original cone, the slant height of the frustum is the difference in slant heights. Alternatively, since the height of the frustum is 8 - 3 = 5 inches, and the difference in radii is 5 - 15/8 = (40/8 - 15/8) = 25/8 inches. So, the slant height s of the frustum is sqrt(5^2 + (25/8)^2).Let me compute that:5^2 is 25, and (25/8)^2 is 625/64. So, 25 is 1600/64, so 1600/64 + 625/64 = 2225/64. Therefore, s = sqrt(2225/64) = sqrt(2225)/8. Let me see if 2225 can be simplified. 2225 divided by 25 is 89, so sqrt(2225) = 5*sqrt(89). Therefore, s = (5*sqrt(89))/8 inches.Now, the lateral surface area of the frustum F is π*(R + r)*s = π*(5 + 15/8)*(5*sqrt(89)/8). Let's compute R + r: 5 is 40/8, so 40/8 + 15/8 = 55/8. Therefore, the lateral surface area of F is π*(55/8)*(5*sqrt(89)/8) = π*(275*sqrt(89))/64.Wait, hold on, that seems a bit off. Let me double-check. The formula is π*(R + r)*s, where R is the larger radius (5 inches) and r is the smaller radius (15/8 inches). So, R + r = 5 + 15/8 = 40/8 + 15/8 = 55/8. The slant height s is 5*sqrt(89)/8. So, multiplying them together: (55/8)*(5*sqrt(89)/8) = (275*sqrt(89))/64. So, yes, that's correct.So, the lateral surface area of F is (275*sqrt(89)/64)*π.Now, the lateral surface area of C is (45*sqrt(89)/64)*π.Therefore, the ratio k of the painted surface areas of C to F is:k = (45*sqrt(89)/64) / (275*sqrt(89)/64) = (45/64) / (275/64) = 45/275.Simplify 45/275: both are divisible by 5. 45 ÷ 5 = 9, 275 ÷ 5 = 55. So, 9/55.Wait, but 9 and 55 have no common factors besides 1, so the ratio k is 9/55. Therefore, m = 9, n = 55, so m + n = 64.Wait, but hold on a second. Let me make sure I didn't make a mistake in calculating the slant height of the frustum. Earlier, I thought of subtracting the slant heights, but actually, since the frustum is part of the original cone, the slant height of the frustum is the difference in slant heights. Let me check that.The original slant height is sqrt(89). The smaller cone's slant height is (3*sqrt(89))/8. So, the slant height of the frustum would be sqrt(89) - (3*sqrt(89))/8 = (8*sqrt(89) - 3*sqrt(89))/8 = (5*sqrt(89))/8. Which matches the earlier calculation. So, that's correct.Alternatively, using the height of the frustum (5 inches) and the difference in radii (25/8 inches), we found the slant height s = sqrt(5^2 + (25/8)^2) = 5*sqrt(89)/8. So, that's consistent.Therefore, the lateral surface area of the frustum is π*(55/8)*(5*sqrt(89)/8) = 275*sqrt(89)/64 * π, which is correct.Similarly, the lateral surface area of C is π*(15/8)*(3*sqrt(89)/8) = 45*sqrt(89)/64 * π.So, the ratio is 45/275, which simplifies to 9/55. Therefore, m = 9, n = 55, so m + n = 64.Wait, but in the initial problem statement, it says "the surface of the cone (excluding the base) is painted." So, when we cut the cone into C and F, does the frustum F have any new painted surfaces? Wait, no, because the original painted surface was the lateral surface. When we cut it, the new surfaces created by the cut are not painted, because the problem says the surface of the cone (excluding the base) is painted. So, the frustum F has its own lateral surface area, which is part of the original painted surface. Similarly, the smaller cone C has its lateral surface area, which is also part of the original painted surface. So, the painted areas are just the lateral surface areas of C and F, respectively.Therefore, my earlier calculation is correct. The ratio is 9/55, so m + n = 64.Wait, but in the initial problem, the user had a different approach and got 2/11, leading to m + n = 13. But I think that approach might have been incorrect because they might have miscalculated the slant heights or the surface areas. Let me double-check my calculations again.Original cone: height 8, radius 5, slant height sqrt(89). Smaller cone: height 3, radius 15/8, slant height (3*sqrt(89))/8. Frustum: height 5, radii 5 and 15/8, slant height 5*sqrt(89)/8.Lateral surface area of C: π*(15/8)*(3*sqrt(89)/8) = 45π*sqrt(89)/64.Lateral surface area of F: π*(5 + 15/8)*(5*sqrt(89)/8) = π*(55/8)*(5*sqrt(89)/8) = 275π*sqrt(89)/64.So, ratio k = (45/64)/(275/64) = 45/275 = 9/55.Yes, that seems correct. So, the ratio is 9/55, so m + n = 64.Wait, but in the initial problem, the user had a different approach and got 2/11. Maybe they made a mistake in calculating the slant height of the frustum. Let me see.In the initial problem, the user calculated the slant height of the frustum as sqrt(5^2 + (25/8)^2) = sqrt(25 + 625/64) = sqrt(2240/64) = sqrt(35). Wait, that can't be right because 2240/64 is 35, so sqrt(35). But 2240 divided by 64 is indeed 35. So, sqrt(35). But wait, that contradicts my earlier calculation where I got 5*sqrt(89)/8.Wait, let's see: 5^2 is 25, and (25/8)^2 is 625/64. So, 25 is 1600/64, so 1600/64 + 625/64 = 2225/64, not 2240/64. So, the user made a mistake here. They wrote 2240/64, but it should be 2225/64. Therefore, sqrt(2225/64) = sqrt(2225)/8 = 5*sqrt(89)/8, as I had before. So, the user incorrectly wrote 2240 instead of 2225, leading to an incorrect slant height of sqrt(35) instead of 5*sqrt(89)/8.Therefore, their calculation of the lateral surface area of F was incorrect, leading to an incorrect ratio. So, my calculation is correct, and the ratio is 9/55, so m + n = 64.Wait, but let me confirm the slant height again. The height of the frustum is 5 inches, and the difference in radii is 5 - 15/8 = 25/8 inches. So, the slant height s = sqrt(5^2 + (25/8)^2) = sqrt(25 + 625/64). Let's compute 25 as 1600/64, so 1600/64 + 625/64 = 2225/64. So, s = sqrt(2225)/8. Now, 2225 factors into 25*89, so sqrt(2225) = 5*sqrt(89). Therefore, s = 5*sqrt(89)/8. So, that's correct.Therefore, the lateral surface area of F is π*(5 + 15/8)*(5*sqrt(89)/8) = π*(55/8)*(5*sqrt(89)/8) = 275π*sqrt(89)/64.Similarly, the lateral surface area of C is π*(15/8)*(3*sqrt(89)/8) = 45π*sqrt(89)/64.So, the ratio is 45/275 = 9/55, so m + n = 64.Therefore, the correct answer is 64.

💡Okay, so I have this function f(x) = e^x - (1/2)x² - x - 1. I need to do three parts: first, find the monotonic intervals and extreme values of its derivative f'(x). Second, show that f(x) is greater than g(x) for x > 0, where g(x) is the function symmetric to f(x) with respect to the origin. Third, prove that if x₁ ≠ x₂ and f(x₁) + f(x₂) = 0, then x₁ + x₂ < 0.Starting with part (I). I need to find f'(x). So, let's differentiate f(x). The derivative of e^x is e^x, the derivative of -(1/2)x² is -x, the derivative of -x is -1, and the derivative of -1 is 0. So, f'(x) = e^x - x - 1.Now, to find the monotonic intervals and extreme values of f'(x). That means I need to look at the derivative of f'(x), which is f''(x). So, f''(x) is the derivative of e^x - x - 1, which is e^x - 1.To find where f'(x) is increasing or decreasing, I can look at the sign of f''(x). If f''(x) > 0, then f'(x) is increasing; if f''(x) < 0, then f'(x) is decreasing.So, f''(x) = e^x - 1. Let's solve when f''(x) = 0: e^x - 1 = 0 => e^x = 1 => x = 0.So, for x < 0, e^x < 1, so f''(x) < 0, meaning f'(x) is decreasing. For x > 0, e^x > 1, so f''(x) > 0, meaning f'(x) is increasing.Therefore, f'(x) is decreasing on (-∞, 0) and increasing on (0, ∞). So, the function f'(x) has a minimum at x = 0.What's the value of f'(0)? f'(0) = e^0 - 0 - 1 = 1 - 0 - 1 = 0. So, the minimum value is 0 at x = 0.So, for part (I), f'(x) is decreasing on (-∞, 0), increasing on (0, ∞), with a minimum at (0, 0).Moving on to part (II). The graph of y = g(x) is symmetric to y = f(x) with respect to the origin. So, symmetry about the origin means that g(x) = -f(-x). Let me verify that: if you reflect over the origin, you replace x with -x and y with -y, so y = f(x) becomes -y = f(-x), so y = -f(-x). Yes, that's correct.So, g(x) = -f(-x). Let's compute that: f(-x) = e^{-x} - (1/2)(-x)^2 - (-x) - 1 = e^{-x} - (1/2)x² + x - 1. Therefore, g(x) = -f(-x) = -e^{-x} + (1/2)x² - x + 1.Now, I need to prove that f(x) > g(x) for x > 0. So, let's compute f(x) - g(x):f(x) - g(x) = [e^x - (1/2)x² - x - 1] - [-e^{-x} + (1/2)x² - x + 1]Simplify this:= e^x - (1/2)x² - x - 1 + e^{-x} - (1/2)x² + x - 1Combine like terms:e^x + e^{-x} - (1/2)x² - (1/2)x² - x + x - 1 - 1Simplify:e^x + e^{-x} - x² - 2So, f(x) - g(x) = e^x + e^{-x} - x² - 2.Let me denote this as F(x) = e^x + e^{-x} - x² - 2. I need to show that F(x) > 0 for x > 0.Let's analyze F(x). Maybe take its derivative to see its behavior.Compute F'(x):F'(x) = derivative of e^x is e^x, derivative of e^{-x} is -e^{-x}, derivative of -x² is -2x, derivative of -2 is 0.So, F'(x) = e^x - e^{-x} - 2x.Now, let's compute F''(x):F''(x) = derivative of e^x is e^x, derivative of -e^{-x} is e^{-x}, derivative of -2x is -2.So, F''(x) = e^x + e^{-x} - 2.Notice that e^x + e^{-x} is always greater than or equal to 2 for all real x, because by AM-GM inequality, (e^x + e^{-x})/2 ≥ sqrt(e^x * e^{-x}) = 1, so e^x + e^{-x} ≥ 2. Thus, F''(x) = e^x + e^{-x} - 2 ≥ 0, with equality only when x = 0.Therefore, F''(x) ≥ 0 for all x, which means F'(x) is increasing on the entire real line.Now, let's evaluate F'(0):F'(0) = e^0 - e^{0} - 2*0 = 1 - 1 - 0 = 0.Since F'(x) is increasing and F'(0) = 0, then for x > 0, F'(x) > 0, and for x < 0, F'(x) < 0.Therefore, F(x) has a minimum at x = 0.Compute F(0):F(0) = e^0 + e^{0} - 0 - 2 = 1 + 1 - 0 - 2 = 0.So, F(x) has a minimum at x = 0 with F(0) = 0. Since F'(x) > 0 for x > 0, F(x) is increasing on [0, ∞). Therefore, for x > 0, F(x) > F(0) = 0.Hence, f(x) - g(x) = F(x) > 0 for x > 0, which implies f(x) > g(x) for x > 0.Alright, that seems solid.Now, part (III). If x₁ ≠ x₂ and f(x₁) + f(x₂) = 0, prove that x₁ + x₂ < 0.Hmm, okay. Let's think about this.First, from part (I), we know that f'(x) is decreasing on (-∞, 0) and increasing on (0, ∞), with a minimum at x = 0. So, f'(0) = 0.Wait, actually, f'(x) = e^x - x - 1. Let's analyze f'(x):At x = 0, f'(0) = 0.For x < 0, since f'(x) is decreasing, but wait, f'(x) is decreasing on (-∞, 0). Wait, hold on: f''(x) = e^x - 1. For x < 0, e^x < 1, so f''(x) < 0, which means f'(x) is concave down, but in terms of monotonicity, since f''(x) is the derivative of f'(x), f'(x) is decreasing on (-∞, 0) and increasing on (0, ∞).So, f'(x) is decreasing on (-∞, 0), reaches a minimum at x = 0, then increases on (0, ∞).Given that f'(0) = 0, so for x < 0, f'(x) is decreasing, starting from what? As x approaches -∞, e^x approaches 0, so f'(x) approaches -x - 1. As x approaches -∞, -x -1 approaches +∞. So, f'(x) starts at +∞ when x approaches -∞, decreases to f'(0) = 0.Similarly, for x > 0, f'(x) increases from 0 to +∞ as x approaches +∞.Therefore, f'(x) is always non-negative? Wait, f'(x) is decreasing on (-∞, 0) from +∞ to 0, and increasing on (0, ∞) from 0 to +∞. So, f'(x) ≥ 0 for all x. Therefore, f(x) is monotonically increasing on ℝ.Wait, is that correct? Because if f'(x) is always non-negative, then f(x) is non-decreasing. Let me check f'(x):At x = 0, f'(0) = 0. For x < 0, f'(x) is decreasing, but since it starts at +∞ and goes to 0, it's always positive. For x > 0, f'(x) increases from 0 to +∞, so also positive. So, f'(x) ≥ 0 for all x, meaning f(x) is monotonically increasing on ℝ.But wait, let's compute f(0): f(0) = e^0 - 0 - 0 - 1 = 1 - 0 - 0 -1 = 0.So, f(0) = 0, and since f is increasing, for x > 0, f(x) > 0, and for x < 0, f(x) < 0.Given that, if f(x₁) + f(x₂) = 0, and x₁ ≠ x₂, then one of them must be positive and the other negative because f(x) is increasing. So, suppose x₁ < 0 and x₂ > 0.We need to show that x₁ + x₂ < 0.Hmm, how can I approach this? Maybe using the properties of f(x) and its symmetry or something else.Wait, from part (II), we know that for x > 0, f(x) > g(x), where g(x) = -f(-x). So, f(x) > -f(-x) for x > 0.Which implies f(x) + f(-x) > 0 for x > 0.So, if I take x = x₂ > 0, then f(x₂) + f(-x₂) > 0.But in our case, f(x₁) + f(x₂) = 0. If x₁ is negative, let's denote x₁ = -a where a > 0. Then, f(-a) + f(x₂) = 0.But from part (II), f(x₂) + f(-x₂) > 0. So, f(x₂) > -f(-x₂).But in our case, f(-a) = -f(x₂). So, f(x₂) = -f(-a).From part (II), f(x₂) > -f(-x₂). So, -f(-a) > -f(-x₂). Multiply both sides by -1 (inequality flips): f(-a) < f(-x₂).But f is increasing, so f(-a) < f(-x₂) implies that -a < -x₂, which implies a > x₂.But a = -x₁, so -x₁ > x₂ => x₁ + x₂ < 0.Yes, that seems to work.Let me write it step by step:Given f(x₁) + f(x₂) = 0, with x₁ ≠ x₂.Since f is increasing and f(0) = 0, if x₁ < 0, then f(x₁) < 0, and x₂ > 0, so f(x₂) > 0.Thus, f(x₁) = -f(x₂).From part (II), for x > 0, f(x) > -f(-x). So, f(x₂) > -f(-x₂).But f(x₁) = -f(x₂) => f(x₁) < f(-x₂) because f(x₂) > -f(-x₂) => -f(x₂) < f(-x₂).Since f is increasing, f(x₁) < f(-x₂) implies x₁ < -x₂.Therefore, x₁ + x₂ < 0.Hence, proved.So, summarizing:(I) f'(x) is decreasing on (-∞, 0), increasing on (0, ∞), with a minimum at (0, 0).(II) f(x) > g(x) for x > 0, where g(x) is the origin reflection of f(x).(III) If f(x₁) + f(x₂) = 0 with x₁ ≠ x₂, then x₁ + x₂ < 0.**Final Answer**(I) The function ( f'(x) ) is decreasing on ( (-infty, 0) ) and increasing on ( (0, infty) ) with a minimum at ( (0, 0) ). So, the final answer is (boxed{f'(x) text{ has a minimum at } x = 0}).(II) It is proven that ( f(x) > g(x) ) for ( x > 0 ). Thus, the final answer is (boxed{f(x) > g(x) text{ for } x > 0}).(III) It is proven that ( x_1 + x_2 < 0 ). Therefore, the final answer is (boxed{x_1 + x_2 < 0}).

💡Alright, let's tackle this problem step by step. I've got four options here, labeled A through D, each presenting a different algebraic operation. My goal is to figure out which one is correct. I'm a bit nervous because exponents and algebra can sometimes trip me up, but I'll take it slow and think it through carefully.Starting with **Option A**: It says ( x^{-2} cdot x^{4} = x^{8} ). Hmm, okay. I remember that when you multiply two exponents with the same base, you add the exponents. So, ( x^{-2} cdot x^{4} ) should be ( x^{-2 + 4} ), which simplifies to ( x^{2} ). Wait, but the option says it equals ( x^{8} ). That doesn't seem right. Maybe I made a mistake? Let me double-check. No, adding -2 and 4 definitely gives 2, not 8. So, Option A is incorrect.Moving on to **Option B**: ( 3x + 2y = 6xy ). This one looks a bit tricky. I know that you can't just add terms with different variables like that. ( 3x ) and ( 2y ) are separate terms, and adding them doesn't give you a product like ( 6xy ). Unless there's some specific relationship between x and y that's not given here, this equation doesn't hold true. So, I think Option B is also incorrect.Next is **Option C**: ( (x^{-3})^{-2} = x^{6} ). Okay, here we have an exponent raised to another exponent. I recall the rule that when you raise a power to a power, you multiply the exponents. So, ( (x^{-3})^{-2} ) should be ( x^{-3 times -2} ), which is ( x^{6} ). That matches what's given in the option. So, Option C seems correct. But I'll keep going to make sure there isn't another correct option.Finally, **Option D**: ( y^{3} div y^{3} = y ). Dividing exponents with the same base means subtracting the exponents. So, ( y^{3} div y^{3} ) should be ( y^{3 - 3} = y^{0} ). And anything raised to the power of 0 is 1, not y. So, Option D is incorrect.After reviewing all the options, only Option C seems to be correct. I think I've got it!

💡Okay, so I have this problem where I need to find (a^4 + b^4 + c^4) given the sums of (a + b + c), (a^2 + b^2 + c^2), and (a^3 + b^3 + c^3). Let me see how I can approach this step by step.First, I remember that when dealing with sums of powers of variables, there are some useful formulas and identities that can help relate these sums. I think using Newton's identities or symmetric sums might be the way to go here. Let me recall what I know.Given the equations:[a + b + c = 2,][a^2 + b^2 + c^2 = 3,][a^3 + b^3 + c^3 = 6,]I need to find (a^4 + b^4 + c^4).I think the first step is to find the elementary symmetric sums, which are (ab + bc + ca) and (abc). Once I have those, I can use them to find higher power sums.Starting with the first equation, (a + b + c = 2). If I square both sides, I can relate it to the second equation. Let me try that.Squaring (a + b + c):[(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca).]Plugging in the known values:[2^2 = 3 + 2(ab + bc + ca).]So,[4 = 3 + 2(ab + bc + ca).]Subtracting 3 from both sides:[1 = 2(ab + bc + ca).]Therefore,[ab + bc + ca = frac{1}{2}.]Alright, that's the first symmetric sum. Now, I need to find (abc). For that, I can use the identity for the cube of the sum.Cubing (a + b + c):[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a).]Wait, no, that's not quite right. Let me recall the correct expansion.The correct expansion is:[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(a + c)(b + c).]But I think another way to express it is:[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b + c)(ab + bc + ca) - 3abc.]Yes, that seems right. Let me verify that.Yes, the formula is:[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b + c)(ab + bc + ca) - 3abc.]So, plugging in the known values:[2^3 = 6 + 3 times 2 times frac{1}{2} - 3abc.]Calculating each term:[8 = 6 + 3 times 2 times frac{1}{2} - 3abc.]Simplify the middle term:[3 times 2 times frac{1}{2} = 3.]So,[8 = 6 + 3 - 3abc.]Adding 6 and 3:[8 = 9 - 3abc.]Subtracting 9 from both sides:[-1 = -3abc.]Dividing both sides by -3:[abc = frac{1}{3}.]Wait, that doesn't seem right because earlier, in the initial problem statement, when I tried solving it, I got (abc = frac{5}{6}). Hmm, maybe I made a mistake here.Let me double-check the expansion of ((a + b + c)^3). I think I might have confused the formula.Wait, no, the correct expansion is:[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(a + c)(b + c).]But actually, another form is:[(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b + c)(ab + bc + ca) - 3abc.]Yes, that's correct. So plugging in the values:[8 = 6 + 3 times 2 times frac{1}{2} - 3abc.]Calculating:[8 = 6 + 3 - 3abc,]which simplifies to:[8 = 9 - 3abc.]So,[-1 = -3abc implies abc = frac{1}{3}.]But in my initial solution, I had (abc = frac{5}{6}). That must mean I made a mistake in my initial thought process.Wait, maybe I used a different method earlier. Let me see. Alternatively, I can use the identity for (a^3 + b^3 + c^3).I know that:[a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ca) + 3abc.]Wait, that's another form. Let me plug in the values here.Given:[6 = 2^3 - 3 times 2 times frac{1}{2} + 3abc.]Calculating:[6 = 8 - 3 times 2 times frac{1}{2} + 3abc.]Simplify the middle term:[3 times 2 times frac{1}{2} = 3.]So,[6 = 8 - 3 + 3abc.]Simplify:[6 = 5 + 3abc.]Subtracting 5:[1 = 3abc implies abc = frac{1}{3}.]Okay, so this confirms that (abc = frac{1}{3}). So in my initial thought process, I must have made a mistake when I thought (abc = frac{5}{6}). That was incorrect.So, now I have:[a + b + c = 2,][ab + bc + ca = frac{1}{2},][abc = frac{1}{3}.]Now, to find (a^4 + b^4 + c^4), I can use the Newton's identities, which relate power sums to elementary symmetric sums.Newton's identities state that:[p_k = (a + b + c)p_{k-1} - (ab + bc + ca)p_{k-2} + abc p_{k-3},]where (p_k = a^k + b^k + c^k).We already know:[p_1 = 2,][p_2 = 3,][p_3 = 6.]We need to find (p_4).Using the recurrence relation for (k = 4):[p_4 = (a + b + c)p_3 - (ab + bc + ca)p_2 + abc p_1.]Plugging in the known values:[p_4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2.]Calculating each term:[2 times 6 = 12,][frac{1}{2} times 3 = frac{3}{2},][frac{1}{3} times 2 = frac{2}{3}.]So,[p_4 = 12 - frac{3}{2} + frac{2}{3}.]To combine these, let's convert them to a common denominator. The denominators are 1, 2, and 3, so the least common denominator is 6.Converting each term:[12 = frac{72}{6},][frac{3}{2} = frac{9}{6},][frac{2}{3} = frac{4}{6}.]So,[p_4 = frac{72}{6} - frac{9}{6} + frac{4}{6} = frac{72 - 9 + 4}{6} = frac{67}{6}.]Wait, that gives me (p_4 = frac{67}{6}), which is approximately 11.1667. But in my initial thought process, I had gotten (frac{34}{3}), which is approximately 11.3333. Hmm, so there's a discrepancy here.Let me check my calculations again.Starting with:[p_4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2.]Calculating each term:[2 times 6 = 12,][frac{1}{2} times 3 = 1.5,][frac{1}{3} times 2 ≈ 0.6667.]So,[p_4 = 12 - 1.5 + 0.6667 ≈ 12 - 1.5 = 10.5 + 0.6667 ≈ 11.1667.]Which is (frac{67}{6}).But earlier, I thought it was (frac{34}{3}), which is about 11.3333. So, which one is correct?Wait, let me double-check the Newton's identities formula. Maybe I misapplied it.The formula is:[p_k = (a + b + c)p_{k-1} - (ab + bc + ca)p_{k-2} + abc p_{k-3}.]So for (k=4):[p_4 = (a + b + c)p_3 - (ab + bc + ca)p_2 + abc p_1.]Yes, that's correct.Plugging in:[p_4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2.]Calculating:[12 - 1.5 + 0.6667 = 11.1667.]So, (frac{67}{6}) is correct.But in my initial thought process, I had a different result because I incorrectly calculated (abc) as (frac{5}{6}) instead of (frac{1}{3}). So, the correct value should be (frac{67}{6}).Wait, but let me verify this with another method to be sure.Another way to find (a^4 + b^4 + c^4) is to square (a^2 + b^2 + c^2) and then subtract (2(a^2b^2 + b^2c^2 + c^2a^2)).So,[(a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2).]We know (a^2 + b^2 + c^2 = 3), so:[3^2 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2).]Which simplifies to:[9 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2).]So, if I can find (a^2b^2 + b^2c^2 + c^2a^2), I can solve for (a^4 + b^4 + c^4).I recall that:[(ab + bc + ca)^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c).]We know (ab + bc + ca = frac{1}{2}) and (abc = frac{1}{3}), and (a + b + c = 2).So,[left(frac{1}{2}right)^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2 times frac{1}{3} times 2.]Calculating:[frac{1}{4} = a^2b^2 + b^2c^2 + c^2a^2 + frac{4}{3}.]Subtracting (frac{4}{3}) from both sides:[frac{1}{4} - frac{4}{3} = a^2b^2 + b^2c^2 + c^2a^2.]Finding a common denominator for the left side:[frac{3}{12} - frac{16}{12} = -frac{13}{12}.]So,[a^2b^2 + b^2c^2 + c^2a^2 = -frac{13}{12}.]Now, plugging this back into the earlier equation:[9 = a^4 + b^4 + c^4 + 2 times left(-frac{13}{12}right).]Simplifying:[9 = a^4 + b^4 + c^4 - frac{13}{6}.]Adding (frac{13}{6}) to both sides:[9 + frac{13}{6} = a^4 + b^4 + c^4.]Converting 9 to sixths:[frac{54}{6} + frac{13}{6} = frac{67}{6}.]So,[a^4 + b^4 + c^4 = frac{67}{6}.]This matches the result I got using Newton's identities. Therefore, the correct answer should be (frac{67}{6}).Wait a minute, but in my initial thought process, I had (frac{34}{3}), which is approximately 11.3333, whereas (frac{67}{6}) is approximately 11.1667. These are close but not the same. So, which one is correct?Let me check the calculations again. Maybe I made a mistake in one of the methods.Starting with the first method using Newton's identities:[p_4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2 = 12 - 1.5 + 0.6667 = 11.1667 = frac{67}{6}.]Second method:[(a^2 + b^2 + c^2)^2 = 9 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2).]Then,[(ab + bc + ca)^2 = frac{1}{4} = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c) = a^2b^2 + b^2c^2 + c^2a^2 + frac{4}{3}.]So,[a^2b^2 + b^2c^2 + c^2a^2 = frac{1}{4} - frac{4}{3} = -frac{13}{12}.]Then,[9 = a^4 + b^4 + c^4 + 2(-frac{13}{12}) = a^4 + b^4 + c^4 - frac{13}{6}.]So,[a^4 + b^4 + c^4 = 9 + frac{13}{6} = frac{54}{6} + frac{13}{6} = frac{67}{6}.]Both methods give the same result, so I must have made a mistake in my initial thought process where I thought (abc = frac{5}{6}). That was incorrect; the correct value is (abc = frac{1}{3}).Therefore, the correct value of (a^4 + b^4 + c^4) is (frac{67}{6}).Wait, but let me check if I applied the Newton's identities correctly. The formula is:[p_k = (a + b + c)p_{k-1} - (ab + bc + ca)p_{k-2} + abc p_{k-3}.]For (k=4), that would be:[p_4 = (a + b + c)p_3 - (ab + bc + ca)p_2 + abc p_1.]Plugging in:[p_4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2 = 12 - 1.5 + 0.6667 = 11.1667 = frac{67}{6}.]Yes, that's correct.Alternatively, using the generating function approach, since we have the characteristic equation of the roots (a, b, c), which is:[x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc = 0,]which is:[x^3 - 2x^2 + frac{1}{2}x - frac{1}{3} = 0.]Then, for each root (x), we have:[x^3 = 2x^2 - frac{1}{2}x + frac{1}{3}.]Multiplying both sides by (x) to get (x^4):[x^4 = 2x^3 - frac{1}{2}x^2 + frac{1}{3}x.]So, summing over all roots (a, b, c):[a^4 + b^4 + c^4 = 2(a^3 + b^3 + c^3) - frac{1}{2}(a^2 + b^2 + c^2) + frac{1}{3}(a + b + c).]Plugging in the known values:[a^4 + b^4 + c^4 = 2 times 6 - frac{1}{2} times 3 + frac{1}{3} times 2 = 12 - 1.5 + 0.6667 = 11.1667 = frac{67}{6}.]This confirms the result again.So, in my initial thought process, I incorrectly calculated (abc) as (frac{5}{6}), leading to an incorrect result of (frac{34}{3}). The correct value of (abc) is (frac{1}{3}), leading to (a^4 + b^4 + c^4 = frac{67}{6}).Therefore, the correct answer is (frac{67}{6}).

💡First, I recognize that the cistern can be filled in 7 hours without any leaks. This means the filling rate is ( frac{1}{7} ) of the cistern per hour.When there is a leak, it takes 8 hours to fill the cistern. This implies that the effective filling rate, accounting for the leak, is ( frac{1}{8} ) of the cistern per hour.To find the rate at which the leak empties the cistern, I subtract the effective filling rate from the original filling rate:[frac{1}{7} - frac{1}{8} = frac{8}{56} - frac{7}{56} = frac{1}{56}]This means the leak empties ( frac{1}{56} ) of the cistern each hour.Finally, to determine how long it takes for the leak to empty the full cistern, I take the reciprocal of the leak rate:[frac{1}{frac{1}{56}} = 56 text{ hours}]

💡Alright, so I have this problem here about two sets, A and B, and I need to find the number of elements in the intersection of the complement of A with B. Let me try to break this down step by step.First, let's look at set A. It's defined as {x | x² - 5x + 4 ≥ 0}. Okay, so this is a quadratic inequality. I remember that to solve quadratic inequalities, it's helpful to factor them if possible. Let me try factoring x² - 5x + 4.Hmm, x² - 5x + 4. I need two numbers that multiply to 4 and add up to -5. Let's see, -1 and -4? Yeah, that works because (-1) * (-4) = 4 and (-1) + (-4) = -5. So, the quadratic factors as (x - 1)(x - 4) ≥ 0.Now, to solve this inequality, I should find the critical points where the expression equals zero. That happens when x = 1 and x = 4. These points divide the number line into three intervals: (-∞, 1), (1, 4), and (4, ∞). I need to test each interval to see where the product (x - 1)(x - 4) is non-negative (greater than or equal to zero).Let's pick a test point in each interval.1. For (-∞, 1), let's choose x = 0. Plugging in, we get (0 - 1)(0 - 4) = (-1)(-4) = 4, which is positive. So, this interval satisfies the inequality.2. For (1, 4), let's choose x = 2. Plugging in, we get (2 - 1)(2 - 4) = (1)(-2) = -2, which is negative. So, this interval does not satisfy the inequality.3. For (4, ∞), let's choose x = 5. Plugging in, we get (5 - 1)(5 - 4) = (4)(1) = 4, which is positive. So, this interval satisfies the inequality.Also, since the inequality is ≥ 0, we include the endpoints x = 1 and x = 4. Therefore, the solution for set A is x ≤ 1 or x ≥ 4. So, A = (-∞, 1] ∪ [4, ∞).Next, I need to find the complement of A with respect to the real numbers, denoted as ∁ℝA. The complement of A would be all real numbers not in A. Since A is (-∞, 1] ∪ [4, ∞), the complement would be the interval between 1 and 4, not including the endpoints because A includes them. So, ∁ℝA = (1, 4).Now, let's look at set B. It's defined as {x ∈ ℤ | |x - 1| ≤ 2}. Okay, so B consists of integers x such that the absolute value of x - 1 is less than or equal to 2. Let's solve this inequality.|x - 1| ≤ 2 means that x - 1 is between -2 and 2. So, -2 ≤ x - 1 ≤ 2. Adding 1 to all parts of the inequality, we get -1 ≤ x ≤ 3. Therefore, x can be any integer from -1 to 3, inclusive.So, B = {-1, 0, 1, 2, 3}.Now, I need to find the intersection of ∁ℝA and B. Remember, ∁ℝA is (1, 4), which is an open interval from 1 to 4, not including 1 and 4. B is the set {-1, 0, 1, 2, 3}.The intersection of these two sets would be the elements that are in both sets. So, looking at B, which elements are also in (1, 4)? Let's list the elements of B:- -1: Is -1 in (1, 4)? No.- 0: Is 0 in (1, 4)? No.- 1: Is 1 in (1, 4)? No, because it's an open interval, so 1 is not included.- 2: Is 2 in (1, 4)? Yes.- 3: Is 3 in (1, 4)? Yes.So, the intersection is {2, 3}. Therefore, there are 2 elements in the intersection.Wait, let me double-check to make sure I didn't miss anything. ∁ℝA is (1, 4), so it includes all real numbers greater than 1 and less than 4. B is the integers from -1 to 3. So, the integers in B that are greater than 1 and less than 4 are 2 and 3. Yep, that's two elements. I think that's correct.So, the number of elements in (∁ℝA) ∩ B is 2.**Final Answer**boxed{B}