💡Alright, so I've got this problem here: I need to find all the integer values of ( n ) that satisfy two inequalities, ( n + 13 > 15 ) and ( -6n > -18 ). Hmm, okay, let's break this down step by step.First, I'll tackle the first inequality: ( n + 13 > 15 ). To solve for ( n ), I need to isolate it on one side. So, I'll subtract 13 from both sides of the inequality. That gives me:[ n + 13 - 13 > 15 - 13 ][ n > 2 ]Alright, so ( n ) has to be greater than 2. Got it.Now, moving on to the second inequality: ( -6n > -18 ). This one looks a bit trickier because of the negative coefficient in front of ( n ). I remember that when you divide or multiply both sides of an inequality by a negative number, you have to reverse the inequality sign. So, let's divide both sides by -6, but don't forget to flip the inequality sign:[ frac{-6n}{-6} < frac{-18}{-6} ][ n < 3 ]Okay, so ( n ) has to be less than 3.Now, putting both inequalities together, ( n > 2 ) and ( n < 3 ). So, ( n ) has to be greater than 2 but less than 3. But wait, the problem specifies that ( n ) should be an integer. Hmm, integers are whole numbers, so what integers are between 2 and 3? Well, 2 is less than 3, but there's no integer between them. So, does that mean there are no integer values that satisfy both inequalities?Wait a minute, maybe I made a mistake somewhere. Let me double-check my steps.Starting with the first inequality: ( n + 13 > 15 ). Subtracting 13 from both sides gives ( n > 2 ). That seems correct.For the second inequality: ( -6n > -18 ). Dividing both sides by -6 and flipping the inequality gives ( n < 3 ). That also seems right.So, combining ( n > 2 ) and ( n < 3 ), the only possible values for ( n ) are numbers greater than 2 but less than 3. But since ( n ) has to be an integer, and there are no integers between 2 and 3, does that mean there are no solutions?Hmm, that seems a bit odd. Maybe I should consider if the inequalities are strict or not. The first inequality is ( n + 13 > 15 ), which is strict, meaning ( n ) has to be strictly greater than 2. The second inequality is ( -6n > -18 ), which also is strict, meaning ( n ) has to be strictly less than 3.So, if ( n ) has to be strictly greater than 2 and strictly less than 3, and ( n ) has to be an integer, then indeed, there are no integer values that satisfy both conditions simultaneously.But wait, maybe I should check if the inequalities allow for equality. If the inequalities were non-strict, like ( n + 13 geq 15 ) and ( -6n geq -18 ), then ( n ) could be 2 or 3. But in this case, the inequalities are strict, so equality isn't allowed.Therefore, after carefully considering both inequalities and ensuring my steps are correct, I conclude that there are no integer values of ( n ) that satisfy both ( n + 13 > 15 ) and ( -6n > -18 ) simultaneously.

💡Okay, so I have this problem about two curves in the Cartesian coordinate system. Let me try to understand what it's asking and figure out how to approach it step by step.First, the problem mentions curve C₁ with parametric equations x = a cos φ and y = a sin φ, where a > 0 and φ is a parameter. It also says that point A(2, 0) lies on curve C₁. Then, there's another curve C₂ given in polar coordinates as ρ = a cos θ. The origin O is the pole, and the positive x-axis is the polar axis.Part (Ⅰ) asks for the Cartesian equation of curve C₂. Hmm, okay. So, I know that polar coordinates can be converted to Cartesian coordinates using the transformations x = ρ cos θ and y = ρ sin θ. Maybe I can use that to rewrite the polar equation of C₂ in terms of x and y.But before that, I think I need to find the value of 'a' because it's given that point A(2, 0) is on curve C₁. Let me work on that first because it might be useful for both parts.So, for curve C₁, the parametric equations are x = a cos φ and y = a sin φ. Since point A(2, 0) is on C₁, substituting x = 2 and y = 0 into the parametric equations should satisfy them. That gives me:2 = a cos φ 0 = a sin φFrom the second equation, 0 = a sin φ. Since a > 0, this implies that sin φ = 0. The solutions for φ where sin φ = 0 are φ = 0, π, 2π, etc. But let's take φ = 0 for simplicity because that would make cos φ = 1, which would satisfy the first equation 2 = a * 1, so a = 2.Okay, so now I know that a = 2. That means the parametric equations for C₁ are x = 2 cos φ and y = 2 sin φ. So, curve C₁ is a circle with radius 2 centered at the origin because the parametric equations are those of a circle.Now, moving on to curve C₂, which is given in polar coordinates as ρ = a cos θ. Since I found that a = 2, the equation becomes ρ = 2 cos θ. I need to convert this into a Cartesian equation.I remember that in polar coordinates, ρ = 2 cos θ can be converted to Cartesian coordinates by multiplying both sides by ρ. So, ρ² = 2 ρ cos θ. But ρ² is x² + y², and ρ cos θ is x. So substituting these in, I get:x² + y² = 2xHmm, that looks like the equation of a circle. Let me rearrange it to standard form. If I move the 2x to the left side:x² - 2x + y² = 0To complete the square for the x terms, I take the coefficient of x, which is -2, divide by 2 to get -1, square it to get 1, and add it to both sides:x² - 2x + 1 + y² = 1 (x - 1)² + y² = 1So, the Cartesian equation of curve C₂ is a circle with center at (1, 0) and radius 1. That makes sense because the polar equation ρ = 2 cos θ is a circle tangent to the origin with diameter along the x-axis.Alright, that takes care of part (Ⅰ). Now, part (Ⅱ) is a bit more involved. It says that the polar coordinates of points M and N are (ρ₁, θ) and (ρ₂, θ + π/2) respectively, and both points lie on curve C₁. I need to calculate the value of 1/ρ₁² + 1/ρ₂².Let me break this down. Points M and N are on curve C₁, which is the circle x² + y² = 4 in Cartesian coordinates because a = 2. Their polar coordinates are given, so I can write their Cartesian coordinates using the polar to Cartesian transformations:For point M: x = ρ₁ cos θ, y = ρ₁ sin θ For point N: x = ρ₂ cos(θ + π/2), y = ρ₂ sin(θ + π/2)I know that cos(θ + π/2) is equal to -sin θ, and sin(θ + π/2) is equal to cos θ. So, the coordinates for point N simplify to:x = -ρ₂ sin θ y = ρ₂ cos θNow, since both points M and N lie on curve C₁, their coordinates must satisfy the equation x² + y² = 4.Let me substitute the coordinates of M into the equation:(ρ₁ cos θ)² + (ρ₁ sin θ)² = 4 ρ₁² (cos² θ + sin² θ) = 4 But cos² θ + sin² θ = 1, so this simplifies to ρ₁² = 4 Therefore, 1/ρ₁² = 1/4Wait, that seems too straightforward. Let me check for point N as well.Substituting the coordinates of N into the equation:(-ρ₂ sin θ)² + (ρ₂ cos θ)² = 4 ρ₂² sin² θ + ρ₂² cos² θ = 4 ρ₂² (sin² θ + cos² θ) = 4 Again, sin² θ + cos² θ = 1, so ρ₂² = 4 Therefore, 1/ρ₂² = 1/4So, adding them together: 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2Wait, that can't be right because the answer seems too simple, and I recall that in the initial problem statement, the parametric equations for C₁ were given as x = a cos φ and y = a sin φ, which is a circle of radius a, but in the thought process above, I considered it as x² + y² = 4, which is correct because a = 2.But let me think again. Maybe I made a mistake in the substitution. Let me verify.For point M: (ρ₁ cos θ, ρ₁ sin θ) lies on C₁, so (ρ₁ cos θ)^2 + (ρ₁ sin θ)^2 = 4 Which is ρ₁² (cos² θ + sin² θ) = 4 Which simplifies to ρ₁² = 4, so 1/ρ₁² = 1/4For point N: (ρ₂ cos(θ + π/2), ρ₂ sin(θ + π/2)) Which is (-ρ₂ sin θ, ρ₂ cos θ) Substituting into C₁: (-ρ₂ sin θ)^2 + (ρ₂ cos θ)^2 = 4 Which is ρ₂² sin² θ + ρ₂² cos² θ = 4 Again, ρ₂² (sin² θ + cos² θ) = 4 So, ρ₂² = 4, hence 1/ρ₂² = 1/4So, adding them: 1/4 + 1/4 = 1/2But wait, in the initial problem, part (Ⅱ) says "calculate the value of 1/ρ₁² + 1/ρ₂²." According to my calculations, it's 1/2. However, I have a feeling that the answer might be different because sometimes when dealing with parametric equations and polar coordinates, there might be more to it.Let me think again. Maybe I misapplied the parametric equations. The parametric equations for C₁ are x = a cos φ, y = a sin φ, which is indeed a circle of radius a. Since a = 2, it's a circle of radius 2. So, any point on C₁ must satisfy x² + y² = 4.But points M and N are given in polar coordinates. For point M, (ρ₁, θ), so in Cartesian, it's (ρ₁ cos θ, ρ₁ sin θ). For point N, (ρ₂, θ + π/2), which is (ρ₂ cos(θ + π/2), ρ₂ sin(θ + π/2)) = (-ρ₂ sin θ, ρ₂ cos θ). Both of these points must lie on C₁, so their coordinates must satisfy x² + y² = 4.So, for M: (ρ₁ cos θ)^2 + (ρ₁ sin θ)^2 = 4 ⇒ ρ₁² = 4 ⇒ 1/ρ₁² = 1/4 For N: (-ρ₂ sin θ)^2 + (ρ₂ cos θ)^2 = 4 ⇒ ρ₂² = 4 ⇒ 1/ρ₂² = 1/4 Thus, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2Hmm, that seems consistent. But let me check if there's another way to approach this problem, maybe using the parametric form directly.Given that points M and N lie on C₁, which is parameterized by φ. So, for point M, we can write:x = 2 cos φ₁ y = 2 sin φ₁But in polar coordinates, x = ρ₁ cos θ and y = ρ₁ sin θ. So, equating these:2 cos φ₁ = ρ₁ cos θ 2 sin φ₁ = ρ₁ sin θSimilarly, for point N, which is at angle θ + π/2:x = 2 cos φ₂ y = 2 sin φ₂But in polar coordinates, x = ρ₂ cos(θ + π/2) = -ρ₂ sin θ y = ρ₂ sin(θ + π/2) = ρ₂ cos θSo, equating these:2 cos φ₂ = -ρ₂ sin θ 2 sin φ₂ = ρ₂ cos θNow, let's square and add the equations for point M:(2 cos φ₁)^2 + (2 sin φ₁)^2 = (ρ₁ cos θ)^2 + (ρ₁ sin θ)^2 4 (cos² φ₁ + sin² φ₁) = ρ₁² (cos² θ + sin² θ) 4 = ρ₁² So, ρ₁² = 4 ⇒ 1/ρ₁² = 1/4Similarly, for point N:(2 cos φ₂)^2 + (2 sin φ₂)^2 = (-ρ₂ sin θ)^2 + (ρ₂ cos θ)^2 4 (cos² φ₂ + sin² φ₂) = ρ₂² (sin² θ + cos² θ) 4 = ρ₂² So, ρ₂² = 4 ⇒ 1/ρ₂² = 1/4Adding them together: 1/4 + 1/4 = 1/2So, it seems that regardless of the approach, the result is 1/2. But wait, I have a doubt because sometimes in these problems, especially when dealing with angles shifted by π/2, there might be a relation between φ₁ and φ₂ that could affect the result. Let me check that.From the equations for point M:2 cos φ₁ = ρ₁ cos θ 2 sin φ₁ = ρ₁ sin θDividing the second equation by the first:(2 sin φ₁)/(2 cos φ₁) = (ρ₁ sin θ)/(ρ₁ cos θ) tan φ₁ = tan θ So, φ₁ = θ + kπ, where k is an integer. Since φ is a parameter, we can take φ₁ = θ.Similarly, for point N:2 cos φ₂ = -ρ₂ sin θ 2 sin φ₂ = ρ₂ cos θDividing the second equation by the first:(2 sin φ₂)/(2 cos φ₂) = (ρ₂ cos θ)/(-ρ₂ sin θ) tan φ₂ = -cot θ tan φ₂ = - (cos θ / sin θ) tan φ₂ = -cot θ = tan(-θ - π/2) So, φ₂ = -θ - π/2 + kπBut since φ is a parameter, we can adjust it accordingly. However, regardless of the specific φ values, when we square and add the equations, we still get ρ₁² = 4 and ρ₂² = 4, leading to 1/ρ₁² + 1/ρ₂² = 1/2.Wait, but I think I might have made a mistake in the initial assumption. Let me think again. The parametric equations for C₁ are x = 2 cos φ, y = 2 sin φ, which is a circle of radius 2. So, any point on C₁ must satisfy x² + y² = 4. Therefore, for any point on C₁, whether in Cartesian or polar coordinates, the distance from the origin is 2. So, ρ = 2 for any point on C₁. But that can't be right because in polar coordinates, ρ varies depending on θ.Wait, no. In polar coordinates, the equation of C₁ would be ρ = 2, because it's a circle of radius 2 centered at the origin. So, any point on C₁ has ρ = 2, regardless of θ. Therefore, ρ₁ = 2 and ρ₂ = 2, so 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But wait, that seems conflicting with the initial problem statement because curve C₂ is given as ρ = 2 cos θ, which is a different circle. So, points M and N are on C₁, which is ρ = 2, but their polar coordinates are given as (ρ₁, θ) and (ρ₂, θ + π/2). So, if C₁ is ρ = 2, then ρ₁ = 2 and ρ₂ = 2, so 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But let me think again. Maybe I'm confusing the curves. Curve C₁ is given parametrically as x = 2 cos φ, y = 2 sin φ, which is indeed ρ = 2 in polar coordinates. So, any point on C₁ has ρ = 2. Therefore, regardless of θ, ρ₁ and ρ₂ must both be 2. Hence, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But wait, in the initial problem, part (Ⅱ) says that points M and N lie on curve C₁, which is the circle of radius 2. So, their ρ values must be 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But I'm a bit confused because in the initial thought process, I thought that the answer might be different, but all the calculations point to 1/2. Maybe I was overcomplicating it.Alternatively, perhaps the problem is expecting a different approach, considering that curve C₂ is ρ = 2 cos θ, which is a circle of radius 1 centered at (1, 0). But in part (Ⅱ), we're dealing with points on C₁, not C₂. So, their ρ values are fixed at 2.Wait, but let me think about the parametric equations again. For curve C₁, x = 2 cos φ, y = 2 sin φ. So, in polar coordinates, ρ = 2, as I said. Therefore, any point on C₁ has ρ = 2, regardless of θ. So, points M and N, being on C₁, must have ρ₁ = 2 and ρ₂ = 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But let me check if there's a different interpretation. Maybe the parametric equations are not for a circle, but for an ellipse? Wait, no, because x = a cos φ and y = a sin φ is a circle when a is the same for both. If a were different, it would be an ellipse, but here a is the same, so it's a circle.Alternatively, maybe I'm misapplying the parametric equations. Let me think again. For curve C₁, x = a cos φ, y = a sin φ, which is a circle of radius a. Since point A(2, 0) is on C₁, substituting x = 2, y = 0 gives 2 = a cos φ and 0 = a sin φ. As before, this implies a = 2 and φ = 0. So, C₁ is indeed a circle of radius 2.Therefore, any point on C₁ must satisfy ρ = 2 in polar coordinates. So, points M and N, being on C₁, must have ρ₁ = 2 and ρ₂ = 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But wait, in the initial problem, part (Ⅱ) says that points M and N are on C₁, but their polar coordinates are (ρ₁, θ) and (ρ₂, θ + π/2). So, their angles are θ and θ + π/2, but their ρ values are determined by their positions on C₁, which is ρ = 2. Therefore, ρ₁ = 2 and ρ₂ = 2, so 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.But I'm still a bit uncertain because sometimes in these problems, especially when dealing with angles shifted by π/2, there might be a relation between the coordinates that could lead to a different result. Let me think about it differently.Suppose I don't assume that ρ₁ and ρ₂ are both 2, but instead express them in terms of θ and then find 1/ρ₁² + 1/ρ₂².Given that point M is (ρ₁, θ) on C₁, which is ρ = 2. So, ρ₁ = 2. Similarly, point N is (ρ₂, θ + π/2) on C₁, so ρ₂ = 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Alternatively, if I didn't know that C₁ is ρ = 2, I could derive it from the parametric equations. Since x = 2 cos φ and y = 2 sin φ, then x² + y² = 4 cos² φ + 4 sin² φ = 4 (cos² φ + sin² φ) = 4. So, x² + y² = 4, which in polar coordinates is ρ² = 4 ⇒ ρ = 2.Therefore, regardless of the angle θ, ρ must be 2 for points on C₁. Hence, ρ₁ = 2 and ρ₂ = 2, leading to 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Wait, but I'm still a bit concerned because sometimes when dealing with angles shifted by π/2, the coordinates might not both lie on the same circle, but in this case, they do because C₁ is a circle centered at the origin with radius 2. So, any point at angle θ or θ + π/2 will still be at a distance of 2 from the origin.Therefore, I think the correct answer is 1/2.But wait, let me check the initial problem again. It says that curve C₂ is given by ρ = a cos θ, which we converted to (x - 1)^2 + y^2 = 1. But in part (Ⅱ), we're dealing with points on C₁, not C₂. So, the value of a is 2, as determined from point A(2, 0) on C₁.Therefore, the calculations for part (Ⅱ) are correct, and the answer is 1/2.But wait, I just realized something. In the initial problem, part (Ⅱ) says that points M and N lie on curve C₁, which is the circle of radius 2. So, their ρ values are 2, but their angles are θ and θ + π/2. However, in the parametric equations, the angle φ is a parameter, not necessarily related to θ. So, perhaps I need to express ρ₁ and ρ₂ in terms of θ and then find 1/ρ₁² + 1/ρ₂².Wait, but since C₁ is ρ = 2, regardless of θ, ρ₁ and ρ₂ must both be 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Alternatively, if I consider that points M and N are on C₁, which is x² + y² = 4, and their polar coordinates are (ρ₁, θ) and (ρ₂, θ + π/2), then substituting into x² + y² = 4 gives ρ₁² = 4 and ρ₂² = 4, so again, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Therefore, I think the answer is indeed 1/2.But wait, I'm going to double-check by considering specific angles. Let's say θ = 0. Then, point M is (ρ₁, 0), which on C₁ must be (2, 0), so ρ₁ = 2. Point N is (ρ₂, π/2), which on C₁ must be (0, 2), so ρ₂ = 2. Therefore, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Similarly, if θ = π/4, then point M is (ρ₁, π/4). On C₁, this would be (2 cos π/4, 2 sin π/4) = (√2, √2). In polar coordinates, ρ₁ = √( (√2)^2 + (√2)^2 ) = √(2 + 2) = √4 = 2. Similarly, point N is (ρ₂, π/4 + π/2) = (ρ₂, 3π/4). On C₁, this is (2 cos 3π/4, 2 sin 3π/4) = (-√2, √2). In polar coordinates, ρ₂ = √( (-√2)^2 + (√2)^2 ) = √(2 + 2) = √4 = 2. Therefore, again, 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.So, regardless of the angle θ, the result is consistent. Therefore, I can confidently say that the value of 1/ρ₁² + 1/ρ₂² is 1/2.But wait, I just realized that in the initial problem, part (Ⅱ) says "points M and N both lie on curve C₁", which is the circle of radius 2. So, their ρ values are fixed at 2, making 1/ρ₁² + 1/ρ₂² = 1/4 + 1/4 = 1/2.Therefore, the final answer is 1/2.

💡First, I need to simplify the expression inside the cube root: (2^3 cdot 5^6 cdot 7^3).Next, I'll take the cube root of each factor separately:- The cube root of (2^3) is (2^{3/3} = 2).- The cube root of (5^6) is (5^{6/3} = 5^2 = 25).- The cube root of (7^3) is (7^{3/3} = 7).Finally, I'll multiply these results together: (2 cdot 25 cdot 7 = 350).

💡Okay, so I have this problem here about vectors. Let me try to understand it step by step. We have three unit vectors: **a**, **b**, and **d**. The problem says that the dot product of **a** and **b** is zero, and the dot product of **a** and **d** is also zero. That means **a** is perpendicular to both **b** and **d**. Hmm, okay, so **a** is orthogonal to both **b** and **d**. Then, it mentions that the angle between **b** and **d** is π/3, which is 60 degrees. So, **b** and **d** form a 60-degree angle between them. The problem then states that **a** can be expressed as m times the cross product of **b** and **d**, so **a** = m(**b** × **d**). We need to find all possible values of m.Alright, so let me think about this. Since **a** is a unit vector and it's equal to m times the cross product of **b** and **d**, which are also unit vectors, I can probably use the properties of cross products to find m.First, I remember that the cross product of two vectors is a vector that's perpendicular to both of them. So, **b** × **d** is perpendicular to both **b** and **d**, which makes sense because **a** is also perpendicular to both **b** and **d**. So, **a** must be parallel to **b** × **d**. That's why **a** can be written as a scalar multiple of **b** × **d**.Now, since **a** is a unit vector, the magnitude of **a** is 1. On the other hand, the magnitude of **b** × **d** is equal to the product of their magnitudes times the sine of the angle between them. Since **b** and **d** are unit vectors, their magnitudes are 1, and the angle between them is π/3. So, the magnitude of **b** × **d** is 1 * 1 * sin(π/3). I remember that sin(π/3) is √3/2. So, the magnitude of **b** × **d** is √3/2. Therefore, the magnitude of **a** is equal to the absolute value of m times the magnitude of **b** × **d**. Since **a** is a unit vector, its magnitude is 1. So, we have:1 = |m| * (√3/2)To solve for |m|, I can rearrange this equation:|m| = 1 / (√3/2) = 2/√3But 2/√3 can be rationalized by multiplying numerator and denominator by √3:|m| = (2√3)/3So, m can be either positive or negative (2√3)/3 because the absolute value is given. Therefore, the possible values of m are (2√3)/3 and -(2√3)/3.Wait, let me double-check that. If **a** is equal to m(**b** × **d**), then the direction of **a** depends on the sign of m. Since **a** is a unit vector, it can point in the same direction as **b** × **d** or the opposite direction. So, m can be positive or negative, which gives us two possible values.Is there anything else I need to consider? Maybe the right-hand rule? Well, since the problem doesn't specify the orientation, both possibilities are valid. So, yes, m can be both (2√3)/3 and -(2√3)/3.I think that's it. I don't see any mistakes in my reasoning. The key points were recognizing that **a** is orthogonal to both **b** and **d**, hence parallel to their cross product, and then using the magnitude of the cross product to find m.**Final Answer**The possible values of ( m ) are (boxed{frac{2sqrt{3}}{3}, -frac{2sqrt{3}}{3}}).

💡Let's solve the problem step by step.**Given:**- Time taken on a busy day: 30 minutes- Time taken on a quieter day: 18 minutes- Speed increase on a quieter day: 12 miles per hour**Steps:**1. **Convert the times to hours:** - 30 minutes = 0.5 hours - 18 minutes = 0.3 hours2. **Let ( v ) be the usual speed in miles per hour.** - Distance on a busy day: ( d = v times 0.5 ) - Distance on a quieter day: ( d = (v + 12) times 0.3 )3. **Set the distances equal:** [ v times 0.5 = (v + 12) times 0.3 ]4. **Solve for ( v ):** [ 0.5v = 0.3v + 3.6 ] [ 0.2v = 3.6 ] [ v = 18 text{ miles per hour} ]5. **Calculate the distance:** [ d = 18 times 0.5 = 9 text{ miles} ]**Answer:** The distance to the school is (boxed{9}) miles.

💡Okay, so I need to figure out the sum of the capacities of all odd subsets of S₄. Let me start by understanding the problem step by step.First, S₄ is the set {1, 2, 3, 4}. A subset X of S₄ can be any combination of these numbers, including the empty set. The capacity of a subset X is defined as the product of all its elements. If X has only one element, then its capacity is just that element. The capacity of the empty set is 0. Now, a subset is called odd if its capacity is odd, and even if its capacity is even. So, I need to find all subsets of S₄ whose product is odd and then sum up those products.Hmm, okay. So, when is a product odd? Well, in mathematics, a product of numbers is odd if and only if all the numbers being multiplied are odd. If there's even one even number in the product, the entire product becomes even. That makes sense because any even number multiplied by anything else results in an even number.So, for a subset X of S₄ to be odd, it must contain only odd numbers from S₄. Let me list the odd numbers in S₄. They are 1 and 3. So, any subset that includes only 1 and/or 3 will have an odd capacity. Subsets that include 2 or 4, which are even numbers, will have an even capacity.Alright, so the odd subsets of S₄ are all the subsets that consist solely of the elements 1 and 3. Let me list them out:1. The empty set: Capacity is 0. But wait, the capacity of the empty set is defined as 0, which is even. So, the empty set is not an odd subset.2. {1}: Capacity is 1, which is odd.3. {3}: Capacity is 3, which is odd.4. {1, 3}: Capacity is 1*3 = 3, which is odd.Are there any other subsets? Let's see. The other subsets of S₄ include elements 2 and/or 4, which are even, so their capacities will be even. For example:- {2}: Capacity is 2, even.- {4}: Capacity is 4, even.- {1, 2}: Capacity is 1*2 = 2, even.- {1, 4}: Capacity is 1*4 = 4, even.- {2, 3}: Capacity is 2*3 = 6, even.- {2, 4}: Capacity is 2*4 = 8, even.- {3, 4}: Capacity is 3*4 = 12, even.- {1, 2, 3}: Capacity is 1*2*3 = 6, even.- {1, 2, 4}: Capacity is 1*2*4 = 8, even.- {1, 3, 4}: Capacity is 1*3*4 = 12, even.- {2, 3, 4}: Capacity is 2*3*4 = 24, even.- {1, 2, 3, 4}: Capacity is 1*2*3*4 = 24, even.So, indeed, the only odd subsets are {1}, {3}, and {1, 3}. Their capacities are 1, 3, and 3 respectively. Now, to find the sum of the capacities of all odd subsets, I just need to add these up:1 (from {1}) + 3 (from {3}) + 3 (from {1, 3}) = 1 + 3 + 3 = 7.So, the sum of the capacities of all odd subsets of S₄ is 7.Wait, let me double-check to make sure I didn't miss any subsets. The subsets of S₄ are all combinations of 1, 2, 3, 4. The ones that don't include 2 or 4 are only the subsets of {1, 3}, which are:- Empty set: capacity 0 (even)- {1}: 1 (odd)- {3}: 3 (odd)- {1, 3}: 3 (odd)So, yes, only three subsets contribute to the sum, and their capacities add up to 7. I think that's correct.Another way to think about it is using generating functions. The generating function for the subsets of S₄ is (1 + x)(1 + 2x)(1 + 3x)(1 + 4x). To find the sum of the capacities of all odd subsets, we need to consider only the terms where the product is odd, which corresponds to the terms where we don't include any even elements (2 and 4). So, effectively, we can ignore the even elements and only consider the generating function for the odd elements, which is (1 + x)(1 + 3x). Expanding this, we get:(1 + x)(1 + 3x) = 1 + 4x + 3x²But wait, the coefficients here represent the sum of products for subsets of different sizes. The constant term is 1 (empty set, which we don't include since it's even), the coefficient of x is 4, which would correspond to the sum of single-element subsets: 1 + 3 = 4. The coefficient of x² is 3, which is the product of both elements: 1*3 = 3. So, adding these together, 4 + 3 = 7, which matches our earlier result.So, using generating functions also gives us the same answer, which is reassuring.Therefore, I'm confident that the sum of the capacities of all odd subsets of S₄ is 7.**Final Answer**The sum of the capacities of all odd subsets of ( S_4 ) is boxed{7}.