💡First, I need to determine the range of distances Gill could have walked based on her average speed and the duration of her walk.Given that her average speed was between 3 km/h and 4 km/h, and she walked for 5 hours, I can calculate the minimum and maximum distances she could have covered.For the minimum distance:[text{Minimum Distance} = 3 , text{km/h} times 5 , text{hours} = 15 , text{km}]For the maximum distance:[text{Maximum Distance} = 4 , text{km/h} times 5 , text{hours} = 20 , text{km}]So, the distance Gill walked must be between 15 km and 20 km.Now, I will evaluate each option to see which falls within this range.- **Option A: 12 km** 12 km is less than 15 km, so it is not within the range.- **Option B: 14 km** 14 km is also less than 15 km, so it is not within the range.- **Option C: 19 km** 19 km is between 15 km and 20 km, so it is within the range.- **Option D: 24 km** 24 km is greater than 20 km, so it is not within the range.- **Option E: 35 km** 35 km is also greater than 20 km, so it is not within the range.Therefore, the only distance that falls within the calculated range is 19 km.

💡Okay, so I have this problem where the real numbers 2, b, and a form a geometric sequence. I need to determine the number of real roots of the quadratic equation ax² + bx + 1/3 = 0. Hmm, let me try to break this down step by step.First, I remember that in a geometric sequence, each term is multiplied by a common ratio to get the next term. So, if 2, b, a are in a geometric sequence, then b must be 2 multiplied by some common ratio, let's call it r. Similarly, a would be b multiplied by r, which is 2r multiplied by r, so that's 2r². So, I can write:b = 2r a = 2r²Okay, that makes sense. Now, I need to substitute these expressions for a and b into the quadratic equation. So, replacing a with 2r² and b with 2r, the equation becomes:2r²x² + 2rx + 1/3 = 0Now, to find the number of real roots of a quadratic equation, I know I need to look at the discriminant. The discriminant formula is b² - 4ac. If the discriminant is positive, there are two distinct real roots; if it's zero, there's exactly one real root; and if it's negative, there are no real roots.So, let's calculate the discriminant for this equation. Here, a is 2r², b is 2r, and c is 1/3. Plugging these into the discriminant formula:Discriminant = (2r)² - 4 * (2r²) * (1/3)Let me compute each part step by step. First, (2r)² is 4r². Then, 4 * (2r²) * (1/3) is... let's see, 4 times 2 is 8, and 8 times (1/3) is 8/3. So, it's 8/3 * r².So, putting it all together, the discriminant is:4r² - (8/3)r²Hmm, let's combine these terms. To subtract these, they need a common denominator. 4r² is the same as 12/3 r². So, 12/3 r² - 8/3 r² equals (12 - 8)/3 r², which is 4/3 r².So, the discriminant is (4/3) r².Now, since r is a real number, r² is always non-negative. And since 4/3 is positive, the entire discriminant is non-negative. But wait, is it ever zero?Well, the discriminant is zero only if r² is zero, which would mean r = 0. But if r is zero, then b = 2r = 0 and a = 2r² = 0. But in the original problem, the sequence is 2, b, a. If both b and a are zero, then it's not a valid geometric sequence because the terms after 2 would be zero, which isn't allowed in a geometric sequence (since the common ratio can't be zero if we're starting from a non-zero term). So, r can't be zero.Therefore, r² is always positive, making the discriminant (4/3)r² positive. Since the discriminant is positive, the quadratic equation has two distinct real roots.Wait, but let me double-check. Is there any other case where the discriminant could be zero or negative? Well, since r is real, and r² is always non-negative, and multiplied by 4/3, which is positive, the discriminant can't be negative. It's either positive or zero. But as we saw earlier, it can't be zero because that would require r = 0, which isn't allowed in this context.So, yeah, the discriminant is always positive, meaning the quadratic equation has two distinct real roots. Therefore, the number of real roots is two.I think that's it. I don't see any mistakes in my reasoning. The key was recognizing the geometric sequence and expressing a and b in terms of the common ratio r, then substituting into the quadratic equation and calculating the discriminant to determine the number of real roots.

💡Okay, so I've got this problem about Danny sitting on a rectangular box. Let me try to visualize it first. A rectangular box has three dimensions: length, width, and height. Let's denote them as l, w, and h respectively.The problem says a few things:1. The area of the front face is half the area of the top face.2. The area of the top face is some multiple of the area of the side face.3. The volume of the box is 192.4. The area of the side face is approximately 32.And the question is asking for the ratio of the area of the top face to the side face.Alright, let's break this down step by step.First, let's recall what each face of the box represents in terms of dimensions:- The front face would typically be the face formed by the width and height, so its area is w * h.- The top face is formed by the length and width, so its area is l * w.- The side face is formed by the length and height, so its area is l * h.Now, according to the problem:1. The area of the front face (w * h) is half the area of the top face (l * w). So, mathematically, that would be: w * h = (1/2) * l * w Hmm, I can simplify this equation by dividing both sides by w (assuming w ≠ 0, which makes sense for a box). That gives: h = (1/2) * l So, the height is half the length. That's useful information.2. The area of the top face (l * w) is some multiple of the area of the side face (l * h). Let's denote this multiple as k. So, l * w = k * (l * h) Again, I can simplify this by dividing both sides by l (assuming l ≠ 0): w = k * h So, the width is k times the height. We'll keep this in mind.3. The volume of the box is given as 192. The volume of a rectangular box is l * w * h, so: l * w * h = 1924. The area of the side face (l * h) is approximately 32. So, l * h ≈ 32Alright, let's see what we have so far:From point 1: h = (1/2) * lFrom point 4: l * h ≈ 32Since h = (1/2) * l, we can substitute this into the equation from point 4:l * (1/2) * l ≈ 32Simplify that:(1/2) * l² ≈ 32Multiply both sides by 2:l² ≈ 64Take the square root of both sides:l ≈ 8So, the length is approximately 8 units.Now, since h = (1/2) * l, then h ≈ 4 units.Now, let's use the volume equation from point 3:l * w * h = 192We know l ≈ 8 and h ≈ 4, so:8 * w * 4 = 192Simplify:32 * w = 192Divide both sides by 32:w = 6So, the width is 6 units.Now, let's verify the area of the side face:l * h = 8 * 4 = 32Which matches the given information, so that checks out.Now, let's find the area of the top face:l * w = 8 * 6 = 48And the area of the side face is 32, as given.So, the ratio of the area of the top face to the side face is 48:32.Simplify this ratio by dividing both numbers by 16:48 ÷ 16 = 332 ÷ 16 = 2So, the ratio is 3:2.Just to make sure everything adds up:- Front face area: w * h = 6 * 4 = 24- Top face area: l * w = 8 * 6 = 48- Side face area: l * h = 8 * 4 = 32Check if front face is half of top face: 24 is half of 48? Yes, that's correct.Check the volume: 8 * 6 * 4 = 192. Correct.Check the side face area: 32. Correct.So, all the given conditions are satisfied.Therefore, the ratio of the area of the top face to the side face is 3:2.

💡Alright, so I've got this problem here, and I need to prove that a certain product is divisible by another product. Let me try to unpack what's given and what I need to show.First, the problem states that ( k, m, n ) are negative integers. Hmm, negative integers? That might be a bit tricky, but maybe it's just a way to denote something else, or perhaps it's a typo and they meant positive integers. I'll keep that in mind and see how it plays out.Next, it says that ( m + k + 1 ) is a prime number greater than ( n + 1 ). Okay, so ( m + k + 1 ) is prime, and it's bigger than ( n + 1 ). That seems important. Maybe this prime number will come into play when considering divisibility.Then, we're given ( C_N = s(s + 1) ). Wait, is that ( C_N ) defined as ( s(s + 1) )? That seems a bit unclear. Maybe ( C_N ) is a function of ( N ), but it's defined in terms of ( s ). I wonder if ( s ) is related to ( N ) somehow. Maybe ( s ) is a variable or another parameter. Hmm, this is a bit confusing. Maybe I need to assume ( C_N ) is a function where ( s ) is a function of ( N ). Or perhaps it's a typo, and it should be ( C_N = N(N + 1) ). That would make more sense because ( C_N ) is often used to denote combinations, but ( N(N + 1) ) is also a common expression. Let me assume that ( C_N = N(N + 1) ) for now, unless something contradicts that later.So, if ( C_N = N(N + 1) ), then ( C_{m+1} = (m+1)(m+2) ), ( C_k = k(k + 1) ), and so on. That seems manageable.Now, the product we need to consider is ( (C_{m+1} - C_k)(C_{m,2} - C_k) cdots (C_{m,n} - C_k) ). Wait, hold on, the notation here is a bit unclear. Is it ( C_{m+1} - C_k ), ( C_{m,2} - C_k ), up to ( C_{m,n} - C_k )? Or is it ( C_{m+1} - C_{k} ), ( C_{m+2} - C_{k} ), up to ( C_{m+n} - C_{k} )? The way it's written, it's ( C_{m,2} ), which might mean ( C_{m,2} ) is a combination function, like ( binom{m}{2} ). But earlier, ( C_N ) was defined as ( s(s + 1) ), which doesn't quite align with combinations. Maybe it's a different notation.Wait, if ( C_N = N(N + 1) ), then ( C_{m,i} ) would be ( (m,i)(m,i + 1) ), but that doesn't make much sense. Maybe the commas are just separators, and it's supposed to be ( C_{m+1} - C_k ), ( C_{m+2} - C_k ), up to ( C_{m+n} - C_k ). That would make more sense, especially since the product is written as ( (C_{m+1} - C_k)(C_{m,2} - C_k) cdots (C_{m,n} - C_k) ). So perhaps the second term is ( C_{m+2} - C_k ), and so on up to ( C_{m+n} - C_k ). I'll go with that interpretation.So, the product is ( prod_{i=1}^{n} (C_{m+i} - C_k) ). And we need to show that this product is divisible by ( C_1 cdot C_2 cdots C_n ). So, ( C_1 = 1 cdot 2 = 2 ), ( C_2 = 2 cdot 3 = 6 ), ( C_3 = 3 cdot 4 = 12 ), and so on. So, the product ( C_1 cdot C_2 cdots C_n ) is ( 2 cdot 6 cdot 12 cdots n(n+1) ).Alright, so the goal is to show that ( prod_{i=1}^{n} (C_{m+i} - C_k) ) is divisible by ( prod_{i=1}^{n} C_i ). That is, the product of these differences is a multiple of the product of the ( C_i )'s.Given that ( m + k + 1 ) is a prime number greater than ( n + 1 ), I suspect that this prime might help in showing that certain factors in the product are present, ensuring divisibility.Let me try to compute ( C_{m+i} - C_k ). Since ( C_N = N(N + 1) ), then:( C_{m+i} - C_k = (m+i)(m+i + 1) - k(k + 1) ).Let me expand this:( (m+i)(m+i + 1) = (m+i)^2 + (m+i) ).Similarly, ( k(k + 1) = k^2 + k ).So, subtracting these:( (m+i)^2 + (m+i) - k^2 - k ).This can be rewritten as:( (m+i)^2 - k^2 + (m+i - k) ).Factor the difference of squares:( (m+i - k)(m+i + k) + (m+i - k) ).Factor out ( (m+i - k) ):( (m+i - k)(m+i + k + 1) ).So, ( C_{m+i} - C_k = (m+i - k)(m+i + k + 1) ).That's a useful factorization. So, each term in the product ( prod_{i=1}^{n} (C_{m+i} - C_k) ) can be written as ( (m+i - k)(m+i + k + 1) ).Therefore, the entire product becomes:( prod_{i=1}^{n} (m+i - k)(m+i + k + 1) ).This can be separated into two products:( left( prod_{i=1}^{n} (m+i - k) right) cdot left( prod_{i=1}^{n} (m+i + k + 1) right) ).Let me denote the first product as ( P_1 = prod_{i=1}^{n} (m+i - k) ) and the second as ( P_2 = prod_{i=1}^{n} (m+i + k + 1) ).So, ( P_1 = prod_{i=1}^{n} (m - k + i) ) and ( P_2 = prod_{i=1}^{n} (m + k + 1 + i) ).Notice that ( P_1 ) is a product of consecutive integers starting from ( m - k + 1 ) up to ( m - k + n ). Similarly, ( P_2 ) is a product of consecutive integers starting from ( m + k + 2 ) up to ( m + k + 1 + n ).So, ( P_1 = frac{(m - k + n)!}{(m - k)!} ) and ( P_2 = frac{(m + k + n + 1)!}{(m + k + 1)!} ).Therefore, the original product is:( frac{(m - k + n)!}{(m - k)!} cdot frac{(m + k + n + 1)!}{(m + k + 1)!} ).Now, the product we need to compare it to is ( C_1 cdot C_2 cdots C_n ), which is ( prod_{i=1}^{n} i(i + 1) ).Let me compute ( C_1 cdot C_2 cdots C_n ):( prod_{i=1}^{n} i(i + 1) = prod_{i=1}^{n} i cdot prod_{i=1}^{n} (i + 1) = (n!) cdot frac{(n + 1)!}{1!} = n! cdot (n + 1)! ).So, ( C_1 cdot C_2 cdots C_n = n! cdot (n + 1)! ).Therefore, we need to show that ( frac{(m - k + n)!}{(m - k)!} cdot frac{(m + k + n + 1)!}{(m + k + 1)!} ) is divisible by ( n! cdot (n + 1)! ).In other words, ( n! cdot (n + 1)! ) divides ( frac{(m - k + n)!}{(m - k)!} cdot frac{(m + k + n + 1)!}{(m + k + 1)!} ).To show divisibility, we can think about the prime factors in the numerator and the denominator. Since ( m + k + 1 ) is a prime number greater than ( n + 1 ), it means that ( m + k + 1 ) is a prime that doesn't divide any of the numbers from 1 to ( n + 1 ). Therefore, ( m + k + 1 ) will only appear in the numerator of ( P_2 ), specifically in ( (m + k + n + 1)! ), but not in the denominator ( (m + k + 1)! ). Thus, ( m + k + 1 ) will be a factor in the entire product, which is helpful for divisibility.But I need to think more carefully about how the factorial terms relate to each other. Let's consider the first part ( frac{(m - k + n)!}{(m - k)!} ). This is equal to ( (m - k + 1)(m - k + 2) cdots (m - k + n) ), which is a product of ( n ) consecutive integers. Similarly, ( frac{(m + k + n + 1)!}{(m + k + 1)!} ) is equal to ( (m + k + 2)(m + k + 3) cdots (m + k + n + 1) ), which is also a product of ( n ) consecutive integers.Now, ( n! ) is the product of the first ( n ) positive integers, and ( (n + 1)! ) is the product of the first ( n + 1 ) positive integers. So, ( n! cdot (n + 1)! ) is ( (n!)^2 cdot (n + 1) ).Wait, no, actually, ( n! cdot (n + 1)! = n! cdot (n + 1) cdot n! = (n!)^2 cdot (n + 1) ). Hmm, is that correct? Wait, no, that's not quite right. Let's compute it properly:( n! cdot (n + 1)! = n! cdot (n + 1) cdot n! = (n!)^2 cdot (n + 1) ).Wait, actually, no. ( (n + 1)! = (n + 1) cdot n! ), so ( n! cdot (n + 1)! = n! cdot (n + 1) cdot n! = (n!)^2 cdot (n + 1) ). So yes, that's correct.But in our case, the product ( P_1 cdot P_2 ) is ( frac{(m - k + n)!}{(m - k)!} cdot frac{(m + k + n + 1)!}{(m + k + 1)!} ). We need to see if ( (n!)^2 cdot (n + 1) ) divides this.Alternatively, perhaps we can think of ( P_1 ) and ( P_2 ) in terms of binomial coefficients or something similar.Wait, ( frac{(m - k + n)!}{(m - k)! cdot n!} ) is equal to ( binom{m - k + n}{n} ), which is an integer. Similarly, ( frac{(m + k + n + 1)!}{(m + k + 1)! cdot (n + 1)!} ) is equal to ( binom{m + k + n + 1}{n + 1} ), which is also an integer.Therefore, ( P_1 = binom{m - k + n}{n} cdot n! ) and ( P_2 = binom{m + k + n + 1}{n + 1} cdot (n + 1)! ).Wait, no, actually:( P_1 = frac{(m - k + n)!}{(m - k)!} = binom{m - k + n}{n} cdot n! ).Similarly, ( P_2 = frac{(m + k + n + 1)!}{(m + k + 1)!} = binom{m + k + n + 1}{n + 1} cdot (n + 1)! ).Therefore, the product ( P_1 cdot P_2 ) is:( binom{m - k + n}{n} cdot n! cdot binom{m + k + n + 1}{n + 1} cdot (n + 1)! ).So, ( P_1 cdot P_2 = binom{m - k + n}{n} cdot binom{m + k + n + 1}{n + 1} cdot n! cdot (n + 1)! ).But we need to show that ( n! cdot (n + 1)! ) divides ( P_1 cdot P_2 ). From the above expression, it's clear that ( n! cdot (n + 1)! ) is a factor of ( P_1 cdot P_2 ), because ( P_1 cdot P_2 ) is equal to ( binom{m - k + n}{n} cdot binom{m + k + n + 1}{n + 1} cdot n! cdot (n + 1)! ). Therefore, ( n! cdot (n + 1)! ) divides ( P_1 cdot P_2 ).But wait, that seems too straightforward. Am I missing something? Let me double-check.We have:( P_1 = frac{(m - k + n)!}{(m - k)!} = binom{m - k + n}{n} cdot n! ).Similarly,( P_2 = frac{(m + k + n + 1)!}{(m + k + 1)!} = binom{m + k + n + 1}{n + 1} cdot (n + 1)! ).Therefore,( P_1 cdot P_2 = binom{m - k + n}{n} cdot binom{m + k + n + 1}{n + 1} cdot n! cdot (n + 1)! ).So, ( P_1 cdot P_2 ) is indeed a multiple of ( n! cdot (n + 1)! ), because ( binom{m - k + n}{n} ) and ( binom{m + k + n + 1}{n + 1} ) are integers. Therefore, ( n! cdot (n + 1)! ) divides ( P_1 cdot P_2 ).But wait, the problem states that ( m + k + 1 ) is a prime number greater than ( n + 1 ). How does that play into this?Well, in the expression for ( P_2 ), we have ( binom{m + k + n + 1}{n + 1} ). Since ( m + k + 1 ) is a prime, and it's greater than ( n + 1 ), it means that ( m + k + 1 ) does not divide any of the numbers from 1 to ( n + 1 ). Therefore, when we compute ( binom{m + k + n + 1}{n + 1} ), the prime ( m + k + 1 ) will be present in the numerator but not in the denominator, ensuring that ( m + k + 1 ) is a factor of ( binom{m + k + n + 1}{n + 1} ).This is important because it ensures that the factor ( m + k + 1 ) is present in the product ( P_1 cdot P_2 ), which might not necessarily be present in ( n! cdot (n + 1)! ) since ( m + k + 1 ) is greater than ( n + 1 ). Therefore, the divisibility holds because all the necessary factors, including the prime ( m + k + 1 ), are accounted for in ( P_1 cdot P_2 ).So, putting it all together, we've shown that:1. Each ( C_{m+i} - C_k ) factors into ( (m+i - k)(m+i + k + 1) ).2. The product of these differences becomes ( P_1 cdot P_2 ), which can be expressed in terms of binomial coefficients multiplied by ( n! cdot (n + 1)! ).3. Since ( m + k + 1 ) is a prime greater than ( n + 1 ), it ensures that the necessary factors are present in ( P_1 cdot P_2 ) to cover ( n! cdot (n + 1)! ).Therefore, the product ( (C_{m+1} - C_k)(C_{m+2} - C_k) cdots (C_{m+n} - C_k) ) is indeed divisible by ( C_1 cdot C_2 cdots C_n ).**Final Answer**boxed{(C_{m+1} - C_{k})(C_{m+2} - C_{k}) cdots (C_{m+n} - C_{k}) text{ is divisible by } C_{1} cdot C_{2} cdots C_{n}}}

💡Alright, so I have this problem about a function f(x) which is given as A sin(ωx + π/6). They tell me that A is positive and ω is positive. The function has a maximum value of 2, and its smallest positive period is 2π. I need to find the analytical expression of f(x) for part (I). Then, for part (II), I have another function g(x) which is cos(x) multiplied by f(x), and I need to find its maximum and minimum values in the interval [-π/6, π/4].Starting with part (I). I remember that the general form of a sine function is A sin(Bx + C) + D, where A is the amplitude, B affects the period, C is the phase shift, and D is the vertical shift. In this case, our function is A sin(ωx + π/6), so there's no vertical shift, just amplitude A, angular frequency ω, and a phase shift of π/6.First, they mention that the maximum value of f(x) is 2. Since the sine function oscillates between -1 and 1, the maximum value of A sin(...) would be A. So, if the maximum is 2, that must mean A is 2. That seems straightforward.Next, they say the smallest positive period is 2π. The period of a sine function is given by 2π divided by the angular frequency ω. So, period T = 2π / ω. They say T is 2π, so setting up the equation: 2π = 2π / ω. Solving for ω, we can divide both sides by 2π, which gives 1 = 1 / ω, so ω must be 1. That makes sense.So, plugging A and ω back into the function, f(x) should be 2 sin(x + π/6). So, that's part (I) done.Moving on to part (II). Here, g(x) is defined as cos(x) multiplied by f(x). So, g(x) = cos(x) * f(x). Since we found f(x) in part (I), which is 2 sin(x + π/6), let me write that down: g(x) = cos(x) * 2 sin(x + π/6).I need to find the maximum and minimum values of g(x) in the interval [-π/6, π/4]. Hmm, okay. So, maybe I can simplify g(x) first before trying to find its extrema.Let me try to expand 2 sin(x + π/6). Using the sine addition formula: sin(a + b) = sin a cos b + cos a sin b. So, sin(x + π/6) = sin x cos(π/6) + cos x sin(π/6). I know that cos(π/6) is √3/2 and sin(π/6) is 1/2. So, substituting those in, we get sin(x + π/6) = sin x * (√3/2) + cos x * (1/2). Therefore, multiplying by 2, f(x) becomes 2*(√3/2 sin x + 1/2 cos x) = √3 sin x + cos x.So, f(x) is √3 sin x + cos x. Therefore, g(x) = cos x * (√3 sin x + cos x). Let me write that out: g(x) = √3 sin x cos x + cos² x.Hmm, okay. So, now I have g(x) expressed in terms of sin x cos x and cos² x. Maybe I can use some trigonometric identities to simplify this further.I remember that sin(2x) = 2 sin x cos x, so sin x cos x is (1/2) sin(2x). So, √3 sin x cos x is √3*(1/2) sin(2x) = (√3/2) sin(2x). Also, cos² x can be written using the double-angle identity: cos² x = (1 + cos(2x))/2.So, substituting these into g(x), we get: g(x) = (√3/2) sin(2x) + (1 + cos(2x))/2. Let me write that as: g(x) = (√3/2) sin(2x) + 1/2 + (cos(2x))/2.Hmm, combining the terms with sin(2x) and cos(2x), maybe I can express this as a single sine function with a phase shift. The expression is of the form A sin(2x) + B cos(2x) + C, which can be written as D sin(2x + φ) + C, where D is the amplitude and φ is the phase shift.So, let's consider the terms (√3/2) sin(2x) + (1/2) cos(2x). Let me factor out the coefficients: √3/2 and 1/2. So, the amplitude D would be sqrt[(√3/2)^2 + (1/2)^2] = sqrt[(3/4) + (1/4)] = sqrt[1] = 1. So, D is 1.Now, the phase shift φ can be found using tan φ = (B/A), where A is the coefficient of sin and B is the coefficient of cos. Wait, actually, in the standard form A sin θ + B cos θ, the phase shift φ is given by tan φ = B/A. So, in our case, A is √3/2 and B is 1/2. So, tan φ = (1/2) / (√3/2) = 1/√3. Therefore, φ = π/6.So, the expression (√3/2) sin(2x) + (1/2) cos(2x) can be written as sin(2x + π/6). Therefore, g(x) becomes sin(2x + π/6) + 1/2.So, now we have g(x) = sin(2x + π/6) + 1/2. That's a much simpler expression. Now, to find the maximum and minimum values of g(x) in the interval [-π/6, π/4], we can analyze this function.First, let's consider the range of sin(2x + π/6). The sine function oscillates between -1 and 1, so sin(2x + π/6) ranges from -1 to 1. Therefore, g(x) = sin(2x + π/6) + 1/2 ranges from (-1 + 1/2) to (1 + 1/2), which is from -1/2 to 3/2. However, this is the global range. We need to find the maximum and minimum specifically in the interval [-π/6, π/4].So, let's analyze the behavior of 2x + π/6 as x varies from -π/6 to π/4.When x = -π/6, 2x + π/6 = 2*(-π/6) + π/6 = -π/3 + π/6 = -π/6.When x = π/4, 2x + π/6 = 2*(π/4) + π/6 = π/2 + π/6 = (3π/6 + π/6) = 4π/6 = 2π/3.So, as x moves from -π/6 to π/4, the argument of the sine function, 2x + π/6, moves from -π/6 to 2π/3.So, we need to find the maximum and minimum of sin(θ) where θ ranges from -π/6 to 2π/3.Let me recall the graph of sine function. From θ = -π/6 to θ = 2π/3, the sine function starts at sin(-π/6) = -1/2, increases to sin(π/2) = 1 at θ = π/2, and then decreases to sin(2π/3) = √3/2.So, the maximum value of sin(θ) in this interval is 1, achieved at θ = π/2, and the minimum value is -1/2, achieved at θ = -π/6.Therefore, the maximum value of g(x) is 1 + 1/2 = 3/2, and the minimum value is -1/2 + 1/2 = 0.But wait, let me double-check if θ = π/2 is within the interval. θ goes from -π/6 to 2π/3, which is approximately from -0.523 radians to 2.094 radians. π/2 is approximately 1.5708 radians, which is within that interval. So, yes, the maximum is indeed 3/2.Similarly, θ = -π/6 is the lower bound, so sin(-π/6) = -1/2 is the minimum value of sin(θ) in this interval, leading to g(x) = -1/2 + 1/2 = 0.Therefore, the maximum value of g(x) in the interval is 3/2, and the minimum value is 0.Let me just recap to make sure I didn't miss anything. We started by expressing g(x) in terms of sine and cosine, then used trigonometric identities to simplify it into a single sine function plus a constant. Then, by analyzing the argument of the sine function over the given interval, we determined the maximum and minimum values.I think that's solid. I don't see any mistakes in my reasoning. So, I'm confident with these results.**Final Answer**Ⅰ. The analytical expression of the function is boxed{2sinleft(x + frac{pi}{6}right)}.Ⅱ. The maximum value of ( g(x) ) is boxed{frac{3}{2}} and the minimum value is boxed{0}.

💡Alright, so I have this problem about finding the volume of a cone. Let me try to understand what's given and what I need to find. The problem says that the base of the cone contains a chord of length ( a ) that subtends an arc of ( alpha^circ ). Also, the height of the cone makes an angle ( beta ) with the slant height. Hmm, okay.First, I need to recall some basic properties of cones. The volume of a cone is given by ( V = frac{1}{3} pi r^2 h ), where ( r ) is the radius of the base and ( h ) is the height. So, I need to find ( r ) and ( h ) in terms of the given quantities ( a ), ( alpha ), and ( beta ).Let me start by visualizing the cone. The base is a circle with radius ( r ). There's a chord ( AK ) in this circle with length ( a ), and this chord subtends an arc of ( alpha^circ ). I remember that the length of a chord in a circle is related to the radius and the central angle it subtends. The formula for the length of a chord is ( a = 2r sinleft(frac{alpha}{2}right) ). So, if I can express ( r ) in terms of ( a ) and ( alpha ), that would be helpful.Let me write that down:[ a = 2r sinleft(frac{alpha}{2}right) ]Solving for ( r ):[ r = frac{a}{2 sinleft(frac{alpha}{2}right)} ]Okay, so that gives me the radius ( r ) in terms of ( a ) and ( alpha ). Good.Now, I need to find the height ( h ) of the cone. The problem says that the height makes an angle ( beta ) with the slant height. Let me think about this. The slant height is the distance from the apex of the cone to any point on the circumference of the base, which is also the hypotenuse of the right triangle formed by the height ( h ), radius ( r ), and slant height ( l ).In this right triangle, the height ( h ) and the slant height ( l ) form an angle ( beta ). So, using trigonometry, specifically the cosine function, I can relate ( h ) and ( l ):[ cos(beta) = frac{h}{l} ]Which means:[ h = l cos(beta) ]But I don't know ( l ) yet. However, I also know that in the same right triangle, the radius ( r ) and the slant height ( l ) are related by the sine function:[ sin(beta) = frac{r}{l} ]So,[ l = frac{r}{sin(beta)} ]Now, substituting this back into the expression for ( h ):[ h = frac{r}{sin(beta)} cos(beta) = r cot(beta) ]Since I already have ( r ) in terms of ( a ) and ( alpha ), I can substitute that in:[ h = frac{a}{2 sinleft(frac{alpha}{2}right)} cot(beta) ]Alright, so now I have both ( r ) and ( h ) expressed in terms of ( a ), ( alpha ), and ( beta ). Let me write them together:[ r = frac{a}{2 sinleft(frac{alpha}{2}right)} ][ h = frac{a cot(beta)}{2 sinleft(frac{alpha}{2}right)} ]Now, I can plug these into the volume formula:[ V = frac{1}{3} pi r^2 h ]Substituting ( r ) and ( h ):[ V = frac{1}{3} pi left( frac{a}{2 sinleft(frac{alpha}{2}right)} right)^2 left( frac{a cot(beta)}{2 sinleft(frac{alpha}{2}right)} right) ]Let me simplify this step by step. First, square ( r ):[ left( frac{a}{2 sinleft(frac{alpha}{2}right)} right)^2 = frac{a^2}{4 sin^2left(frac{alpha}{2}right)} ]Then, multiply by ( h ):[ frac{a^2}{4 sin^2left(frac{alpha}{2}right)} times frac{a cot(beta)}{2 sinleft(frac{alpha}{2}right)} = frac{a^3 cot(beta)}{8 sin^3left(frac{alpha}{2}right)} ]Now, multiply by ( frac{1}{3} pi ):[ V = frac{1}{3} pi times frac{a^3 cot(beta)}{8 sin^3left(frac{alpha}{2}right)} = frac{pi a^3 cot(beta)}{24 sin^3left(frac{alpha}{2}right)} ]So, the volume of the cone is:[ V = frac{pi a^3 cot(beta)}{24 sin^3left(frac{alpha}{2}right)} ]Let me double-check my steps to make sure I didn't make any mistakes. Starting from the chord length formula, I correctly solved for ( r ). Then, using the trigonometric relationships in the right triangle formed by ( h ), ( r ), and ( l ), I correctly expressed ( h ) in terms of ( r ) and ( beta ). Substituting ( r ) into ( h ) also seems correct. Plugging both into the volume formula and simplifying step by step, I think I did that correctly too.I don't see any errors in my reasoning, so I believe this is the correct volume of the cone given the provided conditions.