answer：First, I need to find the length of the hypotenuse ( XY ) using the Pythagorean theorem since ( XYZ ) is a right triangle with legs ( XZ = 15 ) and ( YZ = 8 ).[XY = sqrt{XZ^2 + YZ^2} = sqrt{15^2 + 8^2} = sqrt{225 + 64} = sqrt{289} = 17]Next, I'll calculate the area of triangle ( XYZ ):[text{Area} = frac{1}{2} times XZ times YZ = frac{1}{2} times 15 times 8 = 60]Then, I'll determine the semiperimeter ( s ) of the triangle:[s = frac{XZ + YZ + XY}{2} = frac{15 + 8 + 17}{2} = 20]Finally, I'll use the formula for the inradius ( r ):[r = frac{text{Area}}{text{Semiperimeter}} = frac{60}{20} = 3]Therefore, the radius of the inscribed circle is 3.

answer：First, I need to find the magnitude of the vector ( overrightarrow{a} = (12, 5) ). The magnitude is calculated using the formula:[|overrightarrow{a}| = sqrt{12^2 + 5^2} = sqrt{144 + 25} = sqrt{169} = 13]Next, to find the unit vector parallel to ( overrightarrow{a} ), I divide each component of ( overrightarrow{a} ) by its magnitude:[text{Unit vector} = left( frac{12}{13}, frac{5}{13} right)]Since unit vectors can also point in the opposite direction, the other unit vector parallel to ( overrightarrow{a} ) is:[left( -frac{12}{13}, -frac{5}{13} right)]

answer：To determine the original price of the pants before the sale, let's break down the problem step by step.1. **Understand the discounts:** - During the sale, all products are 50% cheaper. This means the price during the sale is 50% of the original price. - On Saturdays, there's an additional 20% discount on the already discounted price. This means the Saturday price is 80% of the sale price.2. **Calculate the effective price on Saturday:** - Let ( P ) be the original price of the pants. - During the sale, the price is ( 0.5P ). - On Saturday, the price becomes ( 0.8 times 0.5P = 0.4P ).3. **Determine the savings:** - Carla saved R 50.40 by buying on Saturday instead of before the sale. - The savings can be expressed as ( P - 0.4P = 0.6P ). - Therefore, ( 0.6P = 50.40 ).4. **Solve for the original price:** - ( P = frac{50.40}{0.6} = 84 ).Thus, the original price of the pants before the sale was R 84.

answer：First, identify the value per share of the more expensive stock, which is 78.Next, calculate the total value of the more expensive stock by multiplying the number of shares (14) by the value per share:14 shares * 78/share = 1092.Determine the value per share of the less expensive stock by dividing the value of the more expensive stock by 2:78/share / 2 = 39/share.Then, calculate the total value of the less expensive stock by multiplying the number of shares (26) by the value per share:26 shares * 39/share = 1014.Finally, add the total values of both stocks to find the total value of Matthew's assets:1092 + 1014 = 2106.

answer：Okay, so I have this problem where I need to find constants (a) and (b) such that when I multiply (a) by the vector (begin{pmatrix} 3 4 end{pmatrix}) and (b) by the vector (begin{pmatrix} -1 6 end{pmatrix}), and then add those two results together, I get the vector (begin{pmatrix} 2 7 end{pmatrix}). Alright, let me write that out as an equation to make it clearer:[a begin{pmatrix} 3 4 end{pmatrix} + b begin{pmatrix} -1 6 end{pmatrix} = begin{pmatrix} 2 7 end{pmatrix}]So, breaking this down into components, I can write two separate equations, one for the top component and one for the bottom component. For the top component:[3a + (-1)b = 2]Which simplifies to:[3a - b = 2 quad text{(Equation 1)}]For the bottom component:[4a + 6b = 7 quad text{(Equation 2)}]Now, I have a system of two equations with two variables, (a) and (b). I need to solve this system to find the values of (a) and (b). I can use either substitution or elimination to solve this system. I think elimination might be straightforward here because if I can manipulate the equations to eliminate one of the variables, it might be easier. Let me try that.Looking at Equation 1:[3a - b = 2]If I multiply the entire equation by 6, I can align the coefficients of (b) in both equations to make elimination possible. Let's do that:[6 times (3a - b) = 6 times 2]Which gives:[18a - 6b = 12 quad text{(Equation 3)}]Now, Equation 2 is:[4a + 6b = 7]If I add Equation 3 and Equation 2 together, the ( -6b ) and ( +6b ) will cancel each other out, leaving me with an equation in terms of (a) only. Let's add them:[18a - 6b + 4a + 6b = 12 + 7]Simplifying the left side:[(18a + 4a) + (-6b + 6b) = 22a + 0 = 22a]And the right side:[12 + 7 = 19]So, I have:[22a = 19]To solve for (a), I divide both sides by 22:[a = frac{19}{22}]Alright, so (a) is (frac{19}{22}). Now, I need to find (b). I can substitute this value of (a) back into either Equation 1 or Equation 2. Let's use Equation 1 because it looks simpler.Substituting (a = frac{19}{22}) into Equation 1:[3left(frac{19}{22}right) - b = 2]First, let's compute (3 times frac{19}{22}):[frac{57}{22} - b = 2]Now, I need to solve for (b). Let's subtract (frac{57}{22}) from both sides:[- b = 2 - frac{57}{22}]But 2 is the same as (frac{44}{22}), so:[- b = frac{44}{22} - frac{57}{22} = frac{-13}{22}]Multiplying both sides by -1 to solve for (b):[b = frac{13}{22}]So, (b) is (frac{13}{22}).Let me double-check my work to make sure I didn't make any mistakes. I'll plug both (a) and (b) back into the original vector equation to verify.First, compute (a times begin{pmatrix} 3 4 end{pmatrix}):[frac{19}{22} times begin{pmatrix} 3 4 end{pmatrix} = begin{pmatrix} frac{57}{22} frac{76}{22} end{pmatrix}]Simplify the fractions:[begin{pmatrix} frac{57}{22} frac{38}{11} end{pmatrix}]Next, compute (b times begin{pmatrix} -1 6 end{pmatrix}):[frac{13}{22} times begin{pmatrix} -1 6 end{pmatrix} = begin{pmatrix} -frac{13}{22} frac{78}{22} end{pmatrix}]Simplify the fractions:[begin{pmatrix} -frac{13}{22} frac{39}{11} end{pmatrix}]Now, add these two vectors together:[begin{pmatrix} frac{57}{22} - frac{13}{22} frac{38}{11} + frac{39}{11} end{pmatrix} = begin{pmatrix} frac{44}{22} frac{77}{11} end{pmatrix}]Simplify:[begin{pmatrix} 2 7 end{pmatrix}]Which matches the right-hand side of the original equation. So, my solutions for (a) and (b) are correct.I think I've got it. The values of (a) and (b) that satisfy the equation are (frac{19}{22}) and (frac{13}{22}) respectively.

answer：Okay, so I have this problem about three identical cylindrical surfaces with radius R. Their axes are mutually perpendicular, and they touch each other pairwise. There are two parts: part (a) asks for the radius of the smallest sphere that touches these cylinders, and part (b) asks for the radius of the largest cylinder that touches the three given cylinders, with its axis passing inside the triangle formed by the points of contact.Starting with part (a). I need to visualize three cylinders with perpendicular axes touching each other. Since their axes are mutually perpendicular, I can imagine them aligned along the x, y, and z-axes in a 3D coordinate system. Each cylinder has radius R, and they touch each other pairwise, meaning each pair of cylinders touches at a point.I think the smallest sphere that touches all three cylinders would be centered at the origin where the axes intersect. The sphere should be tangent to each cylinder. To find the radius of this sphere, I need to determine the distance from the origin to the surface of each cylinder.For a cylinder aligned along the x-axis, its equation is y² + z² = R². Similarly, the other cylinders will have equations x² + z² = R² and x² + y² = R². The distance from the origin to the surface of each cylinder along the direction where the sphere touches it should be equal.Wait, actually, the sphere will touch each cylinder along a circle. The radius of the sphere will be the distance from the origin to the point where the sphere touches the cylinder. For the cylinder along the x-axis, the closest point on the cylinder's surface to the origin is along the y-z plane. The distance from the origin to this point is R, but since the sphere needs to touch the cylinder, the radius of the sphere should be such that it reaches this point.But actually, the sphere is tangent to the cylinder, so the distance from the center of the sphere (origin) to the cylinder's surface is equal to the sphere's radius. For a cylinder, the distance from a point to the surface is the distance from the point to the axis minus the radius. But in this case, the origin is on the axis of each cylinder, so the distance from the origin to the surface is just R. But that would mean the sphere's radius is R, but that can't be because the sphere needs to touch all three cylinders simultaneously.Wait, maybe I'm misunderstanding. If the sphere is tangent to each cylinder, the distance from the origin to the cylinder's surface is equal to the sphere's radius. But since the origin is on the axis of each cylinder, the distance from the origin to the cylinder's surface is R. Therefore, the sphere's radius should be R. But that seems too straightforward, and the problem mentions it's the smallest sphere that touches these cylinders. Maybe I'm missing something.Alternatively, perhaps the sphere is not centered at the origin. Maybe it's somewhere else. But if it's the smallest sphere, it should be centered at the origin because that's the most symmetric point. Hmm.Wait, another approach: consider the three cylinders as x² + y² = R², x² + z² = R², and y² + z² = R². The sphere that touches all three cylinders must satisfy the condition that the distance from the sphere's center to each cylinder is equal to the sphere's radius.If the sphere is centered at (a, b, c), then the distance from (a, b, c) to each cylinder is equal to the sphere's radius, say r. For the cylinder x² + y² = R², the distance from (a, b, c) to this cylinder is |c|, because the distance from a point to a cylinder along the z-axis is the absolute value of the z-coordinate. Similarly, the distance to the cylinder x² + z² = R² is |b|, and the distance to y² + z² = R² is |a|.Since the sphere must be tangent to each cylinder, these distances must equal the sphere's radius r. Therefore, |a| = |b| = |c| = r. So the center of the sphere is at (r, r, r). Now, the sphere must also lie within the space bounded by the cylinders. The sphere's equation is (x - r)² + (y - r)² + (z - r)² = r².But we also need to ensure that the sphere does not intersect the cylinders beyond the tangent points. Wait, actually, the sphere is tangent to each cylinder at one point. For example, for the cylinder x² + y² = R², the tangent point would be where z = r, and x² + y² = R². Similarly, for the other cylinders.But to find r, we can consider the point of tangency. For the cylinder x² + y² = R², the tangent point on the sphere would be (0, 0, r). Plugging this into the sphere's equation: (0 - r)² + (0 - r)² + (r - r)² = r² ⇒ r² + r² + 0 = r² ⇒ 2r² = r² ⇒ r² = 0, which implies r = 0. That can't be right.Wait, maybe I made a mistake. The point of tangency isn't necessarily (0, 0, r). Let's think differently. The sphere is centered at (r, r, r) and has radius r. The point of tangency on the cylinder x² + y² = R² will lie along the line where z = r, since the distance from the center to the cylinder is r along the z-axis. So, the point of tangency is (x, y, r) where x² + y² = R². This point must also lie on the sphere.So, plugging into the sphere's equation: (x - r)² + (y - r)² + (r - r)² = r² ⇒ (x - r)² + (y - r)² = r². But we also have x² + y² = R². Let's expand the sphere's equation: x² - 2rx + r² + y² - 2ry + r² = r² ⇒ x² + y² - 2r(x + y) + 2r² = r² ⇒ x² + y² - 2r(x + y) + r² = 0.But x² + y² = R², so substituting: R² - 2r(x + y) + r² = 0 ⇒ 2r(x + y) = R² + r² ⇒ x + y = (R² + r²)/(2r).Now, we need to find x and y such that x² + y² = R² and x + y = (R² + r²)/(2r). Let me denote S = x + y and P = xy. Then, x² + y² = S² - 2P = R². So, S² - 2P = R² ⇒ P = (S² - R²)/2.But we also have the point (x, y, r) lying on the sphere, which we've already used. I think I need another condition. Maybe the gradient of the sphere and the cylinder at the point of tangency must be parallel.The gradient of the sphere at (x, y, r) is (2(x - r), 2(y - r), 0). The gradient of the cylinder x² + y² = R² at (x, y, r) is (2x, 2y, 0). For the gradients to be parallel, there must exist a scalar λ such that:2(x - r) = λ * 2x 2(y - r) = λ * 2y 0 = λ * 0From the first equation: (x - r) = λx ⇒ λ = (x - r)/x From the second equation: (y - r) = λy ⇒ λ = (y - r)/yTherefore, (x - r)/x = (y - r)/y ⇒ y(x - r) = x(y - r) ⇒ xy - ry = xy - rx ⇒ -ry = -rx ⇒ y = x.So, x = y. Therefore, from x + y = S, we have 2x = S ⇒ x = S/2. Similarly, y = S/2.From x² + y² = R²: 2x² = R² ⇒ x² = R²/2 ⇒ x = R/√2. Therefore, x = y = R/√2.Then, S = x + y = 2R/√2 = R√2.From earlier, S = (R² + r²)/(2r) ⇒ R√2 = (R² + r²)/(2r).Multiply both sides by 2r: 2r * R√2 = R² + r² ⇒ 2R√2 r = R² + r².Let's rearrange: r² - 2R√2 r + R² = 0.This is a quadratic in r: r² - 2R√2 r + R² = 0.Using the quadratic formula: r = [2R√2 ± √(8R² - 4R²)] / 2 = [2R√2 ± √(4R²)] / 2 = [2R√2 ± 2R] / 2 = R√2 ± R.Since the radius must be positive and smaller than R√2 (because the sphere is inside the space bounded by the cylinders), we take the smaller root: r = R√2 - R = R(√2 - 1).So, the radius of the smallest sphere that touches all three cylinders is R(√2 - 1).Now, moving on to part (b). We need to find the radius of the largest cylinder that touches the three given cylinders, with its axis passing inside the triangle formed by the points of contact.First, let's understand the configuration. The three given cylinders touch each other pairwise, so each pair touches at a point. The points of contact form a triangle. The axis of the new cylinder passes inside this triangle.I think the new cylinder will be tangent to each of the three given cylinders. Since the given cylinders have radius R and are mutually perpendicular, their points of contact are at (R, R, 0), (R, 0, R), and (0, R, R). So, the triangle is formed by these three points.The axis of the new cylinder must pass through the interior of this triangle. Since the triangle is in 3D space, the axis will be a line segment inside the triangle. The largest such cylinder would be the one with the maximum possible radius while still being tangent to all three given cylinders.To find the radius, let's consider the geometry. The new cylinder must be tangent to each of the three given cylinders. Let's denote the radius of the new cylinder as r. The distance between the axis of the new cylinder and the axis of each given cylinder must be equal to R + r.But the axes of the given cylinders are along the x, y, and z-axes. The axis of the new cylinder is inside the triangle formed by (R, R, 0), (R, 0, R), and (0, R, R). So, the axis of the new cylinder is a line inside this triangle.Wait, actually, the triangle is in 3D space, so it's not a planar triangle. The points (R, R, 0), (R, 0, R), and (0, R, R) form a triangle in the plane x + y + z = R√2? Wait, no. Let me check: x + y + z for (R, R, 0) is 2R, for (R, 0, R) is 2R, and for (0, R, R) is 2R. So, the triangle lies on the plane x + y + z = 2R.The axis of the new cylinder must pass through this plane. The new cylinder is tangent to each of the three given cylinders, so the distance from the axis of the new cylinder to each of the x, y, and z-axes must be equal to R + r.But the axis of the new cylinder is inside the triangle, so it's somewhere in the plane x + y + z = 2R. Let's parametrize the axis of the new cylinder. Since it's a line, we can represent it parametrically.But maybe a better approach is to consider the symmetry. The problem is symmetric with respect to the three cylinders, so the new cylinder's axis should pass through the centroid of the triangle formed by the points of contact.The centroid of the triangle with vertices (R, R, 0), (R, 0, R), and (0, R, R) is ((R + R + 0)/3, (R + 0 + R)/3, (0 + R + R)/3) = (2R/3, 2R/3, 2R/3).So, the axis of the new cylinder passes through the centroid (2R/3, 2R/3, 2R/3). Since the axis is a line, we need to define its direction. Given the symmetry, the axis should be along the line connecting the origin to the centroid, which is the line x = y = z.Wait, but the centroid is (2R/3, 2R/3, 2R/3), so the line from the origin to the centroid is indeed x = y = z. However, the axis of the new cylinder is inside the triangle, so maybe it's better to consider the axis as passing through the centroid and being perpendicular to the plane of the triangle.Wait, the plane of the triangle is x + y + z = 2R, and the normal vector to this plane is (1, 1, 1). So, the axis of the new cylinder, being perpendicular to the plane, would have direction (1, 1, 1). But the axis is a line, so it's parametrized as (t, t, t) for some t.But the axis must pass through the centroid (2R/3, 2R/3, 2R/3), so t = 2R/3. Therefore, the axis is the line x = y = z = 2R/3 + s(1, 1, 1), where s is a parameter.Wait, no. If the axis is along the direction (1, 1, 1) and passes through (2R/3, 2R/3, 2R/3), then the parametric equations are x = 2R/3 + s, y = 2R/3 + s, z = 2R/3 + s. But this line is actually the same as x = y = z, just shifted by 2R/3.But the distance from this axis to each of the x, y, z-axes must be equal to R + r. Let's compute the distance from the axis to the x-axis.The distance from a line to another line in 3D can be found using the formula involving the cross product. The distance between two skew lines is |(P2 - P1) · (v1 × v2)| / |v1 × v2|, where P1 and P2 are points on each line, and v1 and v2 are their direction vectors.But in this case, the axis of the new cylinder is the line x = y = z = 2R/3 + s, and the x-axis is the line y = 0, z = 0, x = t. These lines are skew.Let me denote the axis of the new cylinder as L1: (2R/3, 2R/3, 2R/3) + s(1, 1, 1), and the x-axis as L2: (t, 0, 0).To find the distance between L1 and L2, we can use the formula. Let P1 = (2R/3, 2R/3, 2R/3) and P2 = (0, 0, 0) (a point on L2). The vector P2 - P1 is (-2R/3, -2R/3, -2R/3). The direction vectors of L1 and L2 are v1 = (1, 1, 1) and v2 = (1, 0, 0).Compute v1 × v2: (1, 1, 1) × (1, 0, 0) = (0*0 - 1*0, 1*1 - 1*0, 1*0 - 1*1) = (0, 1, -1).The magnitude of v1 × v2 is √(0² + 1² + (-1)²) = √2.Now, compute (P2 - P1) · (v1 × v2): (-2R/3, -2R/3, -2R/3) · (0, 1, -1) = 0*(-2R/3) + 1*(-2R/3) + (-1)*(-2R/3) = (-2R/3) + (2R/3) = 0.Wait, that can't be right. If the distance is zero, that would mean the lines intersect, but they don't. I must have made a mistake.Wait, no. The formula gives the distance between two skew lines. If the result is zero, it means the lines are coplanar, but in this case, they are skew. So, perhaps I made a mistake in the calculation.Wait, let's recalculate the cross product. v1 = (1, 1, 1), v2 = (1, 0, 0). The cross product is:i j k1 1 11 0 0= i*(1*0 - 1*0) - j*(1*0 - 1*1) + k*(1*0 - 1*1) = i*(0) - j*(-1) + k*(-1) = (0, 1, -1). So that's correct.Then, (P2 - P1) · (v1 × v2) = (-2R/3, -2R/3, -2R/3) · (0, 1, -1) = 0*(-2R/3) + 1*(-2R/3) + (-1)*(-2R/3) = (-2R/3) + (2R/3) = 0.Hmm, so the distance is zero, which suggests that the lines are coplanar, but they shouldn't be. Wait, maybe I chose the wrong point on L2. Instead of (0, 0, 0), maybe I should choose a different point.Wait, no. The distance between two skew lines is the minimum distance between any two points on the lines. If the dot product is zero, it means that the vector connecting P1 and P2 is perpendicular to the cross product, which is the direction of the common perpendicular. So, the distance is |(P2 - P1) · (v1 × v2)| / |v1 × v2| = |0| / √2 = 0. That can't be right because the lines are skew.Wait, I think I made a mistake in choosing P2. I should choose a point on L2 such that the vector P2 - P1 is along the common perpendicular. Alternatively, maybe I should use a different approach.Alternatively, the distance from the axis of the new cylinder to the x-axis can be found by considering the closest point on the x-axis to the axis of the new cylinder.Let me parametrize the axis of the new cylinder as (2R/3 + s, 2R/3 + s, 2R/3 + s). The x-axis is (t, 0, 0). The distance squared between a point on L1 and a point on L2 is:(2R/3 + s - t)² + (2R/3 + s - 0)² + (2R/3 + s - 0)².To minimize this distance, we can take partial derivatives with respect to s and t and set them to zero.Let D² = (2R/3 + s - t)² + (2R/3 + s)² + (2R/3 + s)².Compute ∂D²/∂t = 2(2R/3 + s - t)(-1) = 0 ⇒ 2R/3 + s - t = 0 ⇒ t = 2R/3 + s.Compute ∂D²/∂s = 2(2R/3 + s - t)(1) + 2(2R/3 + s)(1) + 2(2R/3 + s)(1) = 0.Substitute t = 2R/3 + s into the derivative:2(2R/3 + s - (2R/3 + s))(1) + 2(2R/3 + s)(1) + 2(2R/3 + s)(1) = 0 ⇒ 2(0) + 4(2R/3 + s) = 0 ⇒ 4(2R/3 + s) = 0 ⇒ s = -2R/3.Then, t = 2R/3 + (-2R/3) = 0.So, the closest points are on L1: (2R/3 - 2R/3, 2R/3 - 2R/3, 2R/3 - 2R/3) = (0, 0, 0), and on L2: (0, 0, 0). Wait, that's the origin. But the origin is on both L1 and L2? No, L1 is (2R/3 + s, 2R/3 + s, 2R/3 + s), which when s = -2R/3 gives (0, 0, 0). So, the lines intersect at the origin. But that contradicts the earlier thought that they are skew.Wait, actually, if s = -2R/3, then the point on L1 is (0, 0, 0), which is also on L2. So, the lines intersect at the origin. Therefore, the distance between them is zero, which means they are not skew but intersecting. That's a problem because the axis of the new cylinder is supposed to pass through the centroid, which is (2R/3, 2R/3, 2R/3), and the x-axis is along (t, 0, 0). So, the lines intersect at the origin, which is not inside the triangle.Wait, maybe I made a mistake in defining the axis of the new cylinder. If the axis passes through the centroid and is along the direction (1, 1, 1), it actually intersects the x-axis at the origin, which is outside the triangle. Therefore, this axis is not entirely inside the triangle.So, perhaps the axis of the new cylinder is not along (1, 1, 1), but rather lies within the plane x + y + z = 2R and is perpendicular to it. Wait, but the normal vector to the plane is (1, 1, 1), so a line perpendicular to the plane would have direction (1, 1, 1). But as we saw, such a line would intersect the x-axis at the origin, which is outside the triangle.Alternatively, maybe the axis of the new cylinder is not passing through the centroid but is instead a line segment inside the triangle. Since the triangle is in the plane x + y + z = 2R, the axis must lie within this plane.Let me consider the triangle with vertices (R, R, 0), (R, 0, R), and (0, R, R). The centroid is (2R/3, 2R/3, 2R/3). The axis of the new cylinder is a line segment inside this triangle. To maximize the radius, the axis should be as far as possible from the edges of the triangle, meaning it should pass through the centroid.But how do we find the distance from this axis to each of the x, y, z-axes? Since the axis is inside the triangle, it's not straightforward to compute the distance to the x, y, z-axes.Wait, maybe instead of trying to compute the distance directly, we can use the fact that the new cylinder is tangent to each of the three given cylinders. The distance between the axes of the new cylinder and each given cylinder must be equal to R + r.Since the given cylinders are along the x, y, and z-axes, the distance from the axis of the new cylinder to each of these axes must be R + r.But the axis of the new cylinder is a line in the plane x + y + z = 2R. Let's denote a general point on this axis as (a, b, c) where a + b + c = 2R. The direction vector of the axis is perpendicular to the plane, so it's along (1, 1, 1). Therefore, the axis can be parametrized as (a, b, c) + t(1, 1, 1), where a + b + c = 2R.But the distance from this axis to the x-axis is the minimum distance between the two lines. Similarly for the y and z-axes.Wait, let's consider the distance from the axis of the new cylinder to the x-axis. The x-axis is (t, 0, 0). The axis of the new cylinder is (a, b, c) + t(1, 1, 1). The distance between these two lines can be found using the formula for the distance between two skew lines.The formula is |(P2 - P1) · (v1 × v2)| / |v1 × v2|, where P1 and P2 are points on each line, and v1 and v2 are their direction vectors.Let P1 = (a, b, c) on the new cylinder's axis, and P2 = (0, 0, 0) on the x-axis. The direction vectors are v1 = (1, 1, 1) and v2 = (1, 0, 0).Compute v1 × v2: (1, 1, 1) × (1, 0, 0) = (0, 1, -1).Compute (P2 - P1) · (v1 × v2): (-a, -b, -c) · (0, 1, -1) = 0*(-a) + 1*(-b) + (-1)*(-c) = -b + c.The magnitude of v1 × v2 is √(0² + 1² + (-1)²) = √2.Therefore, the distance is | -b + c | / √2.But since the axis is in the plane x + y + z = 2R, we have a + b + c = 2R. Also, the axis is along (1, 1, 1), so the direction vector is (1, 1, 1). Therefore, the vector from P1 to P2 is (-a, -b, -c), and we have the condition that this vector is perpendicular to the direction vector of the new cylinder's axis.Wait, no. The vector connecting P1 and P2 is (-a, -b, -c). For the distance to be minimal, this vector must be perpendicular to both direction vectors v1 and v2. But since v1 and v2 are not necessarily parallel, this might not hold.Alternatively, since we have the distance formula, we can set it equal to R + r.So, | -b + c | / √2 = R + r.Similarly, the distance from the new cylinder's axis to the y-axis and z-axis will give us similar equations.For the y-axis, the distance would be | -a + c | / √2 = R + r.For the z-axis, the distance would be | -a + b | / √2 = R + r.So, we have three equations:1. | -b + c | / √2 = R + r2. | -a + c | / √2 = R + r3. | -a + b | / √2 = R + rSince the axis is inside the triangle, and the triangle is symmetric, we can assume that a = b = c. But wait, if a = b = c, then a + b + c = 3a = 2R ⇒ a = 2R/3. So, a = b = c = 2R/3.Then, substituting into the equations:1. | -2R/3 + 2R/3 | / √2 = 0 = R + r ⇒ R + r = 0, which is impossible since R and r are positive.Therefore, our assumption that a = b = c is incorrect. Instead, let's consider that the distances are equal, so | -b + c | = | -a + c | = | -a + b |.This implies that -b + c = -a + c ⇒ -b = -a ⇒ a = b.Similarly, -a + c = -a + b ⇒ c = b.Therefore, a = b = c.But as before, this leads to a contradiction because then the distance would be zero. So, perhaps the absolute values are equal but the expressions inside could have different signs.Let me consider that | -b + c | = | -a + c | = | -a + b |.Assume that -b + c = -a + c = -a + b.From -b + c = -a + c ⇒ -b = -a ⇒ a = b.From -a + c = -a + b ⇒ c = b.Therefore, again a = b = c, which leads to the same contradiction.Alternatively, maybe the expressions inside the absolute value are equal in magnitude but opposite in sign.Suppose -b + c = -(-a + c) ⇒ -b + c = a - c ⇒ -b + c = a - c ⇒ a + b = 2c.Similarly, from -a + c = -(-a + b) ⇒ -a + c = a - b ⇒ -a + c = a - b ⇒ 2a = b + c.And from -a + b = -(-b + c) ⇒ -a + b = b - c ⇒ -a = -c ⇒ a = c.So, from a = c, and from a + b = 2c ⇒ a + b = 2a ⇒ b = a.Therefore, a = b = c, which again leads to the same contradiction.This suggests that our initial approach might be flawed. Maybe instead of trying to compute the distance from the axis to the x, y, z-axes, we should consider the distance from the axis to the surfaces of the given cylinders.Wait, the new cylinder is tangent to each of the given cylinders. The distance between the axes of the new cylinder and each given cylinder must be equal to R + r.But the given cylinders have their axes along the x, y, z-axes, and the new cylinder's axis is in the plane x + y + z = 2R. So, the distance from the new cylinder's axis to each of the x, y, z-axes must be R + r.But how do we compute this distance?Alternatively, consider that the new cylinder is tangent to each of the three given cylinders. The point of tangency on each given cylinder will lie along the line connecting the axes of the two cylinders.For example, the point of tangency on the x-axis cylinder will lie along the line connecting the x-axis and the new cylinder's axis. Similarly for the other cylinders.But since the new cylinder's axis is in the plane x + y + z = 2R, the line connecting the x-axis and the new cylinder's axis will have a direction vector that is the difference between a point on the new cylinder's axis and a point on the x-axis.Wait, this is getting too abstract. Maybe a better approach is to consider the coordinates of the points of tangency.Let me denote the new cylinder's axis as passing through the centroid (2R/3, 2R/3, 2R/3) and being perpendicular to the plane x + y + z = 2R, so its direction vector is (1, 1, 1). Therefore, the parametric equation of the axis is (2R/3 + t, 2R/3 + t, 2R/3 + t).Now, the new cylinder has radius r, so the distance from any point on its axis to the surface of the new cylinder is r. The new cylinder is tangent to each of the given cylinders, so the distance from the axis of the new cylinder to the axis of each given cylinder is R + r.But the axes of the given cylinders are along the x, y, z-axes. So, the distance from the new cylinder's axis to the x-axis is R + r.As before, the distance between the lines (2R/3 + t, 2R/3 + t, 2R/3 + t) and (s, 0, 0) is R + r.Using the distance formula between two skew lines:Distance = |(P2 - P1) · (v1 × v2)| / |v1 × v2|Where P1 = (2R/3, 2R/3, 2R/3), P2 = (0, 0, 0), v1 = (1, 1, 1), v2 = (1, 0, 0).Compute v1 × v2 = (1, 1, 1) × (1, 0, 0) = (0, 1, -1).Compute (P2 - P1) · (v1 × v2) = (-2R/3, -2R/3, -2R/3) · (0, 1, -1) = 0*(-2R/3) + 1*(-2R/3) + (-1)*(-2R/3) = (-2R/3) + (2R/3) = 0.Wait, this again gives zero, which suggests the lines are coplanar, but they are skew. This is confusing.Alternatively, maybe the distance is not zero, but I'm miscalculating. Let me double-check.Wait, the formula is |(P2 - P1) · (v1 × v2)| / |v1 × v2|.We have (P2 - P1) = (-2R/3, -2R/3, -2R/3).v1 × v2 = (0, 1, -1).Dot product: (-2R/3)(0) + (-2R/3)(1) + (-2R/3)(-1) = 0 - 2R/3 + 2R/3 = 0.So, the distance is zero, which implies the lines are coplanar. But they are not, because the new cylinder's axis is in the plane x + y + z = 2R, and the x-axis is in the plane z = 0, y = 0, so they are skew.This contradiction suggests that my approach is incorrect. Maybe I need to consider the distance from the axis of the new cylinder to the surface of the given cylinders, not just the distance between the axes.Wait, the distance between the axes is R + r, but the distance from the axis of the new cylinder to the surface of the given cylinder is R + r. But the distance from the axis of the new cylinder to the surface of the given cylinder is the distance from the axis to the axis of the given cylinder minus the radius of the given cylinder.Wait, no. The distance between the axes is R + r, so the distance from the new cylinder's axis to the given cylinder's axis is R + r. Therefore, the distance from the new cylinder's axis to the surface of the given cylinder is R + r - R = r.But that doesn't make sense because the new cylinder's radius is r, so the distance from its axis to the surface is r. Therefore, the distance between the axes must be R + r.So, going back, the distance between the new cylinder's axis and the x-axis is R + r.But earlier, we saw that the distance formula gives zero, which is a contradiction. Therefore, perhaps the axis of the new cylinder is not along (1, 1, 1), but rather lies in the plane x + y + z = 2R and is not intersecting the x-axis.Wait, maybe the axis of the new cylinder is not passing through the centroid but is instead a line segment inside the triangle. Let me consider the triangle in the plane x + y + z = 2R with vertices (R, R, 0), (R, 0, R), and (0, R, R). The largest cylinder that fits inside this triangle and touches all three sides would have its axis along the centroid.But in 3D, it's more complex. Alternatively, maybe the radius of the new cylinder is R√2.Wait, considering the symmetry, if the new cylinder is tangent to all three given cylinders, its radius might be R√2. But let me verify.If the new cylinder has radius R√2, then the distance from its axis to each given cylinder's axis would be R + R√2. But I'm not sure.Alternatively, considering the triangle formed by the points of contact, which is an equilateral triangle with side length R√2. The largest circle that fits inside this triangle (incircle) has radius R√2 / √3. But this is in 2D. In 3D, it's more complex.Wait, perhaps the radius of the new cylinder is R√2. Because the distance from the centroid to each vertex is R√2 / √3, but I'm not sure.Alternatively, maybe the radius is R(√3 - 1). But I need a better approach.Let me consider the coordinates. The points of contact are (R, R, 0), (R, 0, R), and (0, R, R). The centroid is (2R/3, 2R/3, 2R/3). The axis of the new cylinder must pass through this centroid and be perpendicular to the plane, so direction (1, 1, 1).The distance from this axis to the x-axis is R + r. As before, using the distance formula, we get zero, which is a problem. Therefore, perhaps the axis is not along (1, 1, 1), but rather lies in the plane and is perpendicular to the plane's normal.Wait, the normal vector is (1, 1, 1), so a line perpendicular to the plane would have direction (1, 1, 1). But as we saw, this leads to the axis intersecting the x-axis at the origin, which is outside the triangle.Therefore, maybe the axis is not perpendicular to the plane but lies within the plane. Let me consider the axis as a line in the plane x + y + z = 2R, not necessarily perpendicular to it.In this case, the direction vector of the axis can be arbitrary, but to maximize the radius, it should be as far as possible from the edges of the triangle.Wait, perhaps the axis is the line segment connecting the midpoints of the edges of the triangle. The midpoints are (R, R/2, R/2), (R/2, R, R/2), and (R/2, R/2, R). The centroid is (2R/3, 2R/3, 2R/3), which is the average of these midpoints.But I'm not sure how this helps.Alternatively, consider that the new cylinder must be tangent to each of the three given cylinders. The points of tangency will lie on the lines where the new cylinder's surface meets the given cylinders.For example, the point of tangency on the x-axis cylinder will lie on the line where the new cylinder's surface intersects the x-axis cylinder. Similarly for the other cylinders.But this is getting too vague. Maybe a better approach is to use coordinates.Let me assume that the new cylinder's axis is along the line x = y = z = t, but shifted so that it's inside the triangle. Wait, but earlier we saw that this line intersects the x-axis at the origin, which is outside the triangle.Alternatively, maybe the axis is a line segment from (R, R, 0) to (R, 0, R) to (0, R, R), but that's the triangle itself, not a line.Wait, perhaps the axis is the line connecting the midpoints of the edges of the triangle. The midpoints are (R, R/2, R/2), (R/2, R, R/2), and (R/2, R/2, R). The line connecting these midpoints is the medial axis of the triangle.But the medial axis in 3D is more complex. Alternatively, the largest cylinder that fits inside the triangle would have its axis along the line that is equidistant from all three edges of the triangle.In 2D, the largest circle that fits inside a triangle is the incircle, with radius r = A/s, where A is the area and s is the semiperimeter.In 3D, for a triangle, the largest cylinder that fits inside would have its axis along the line that is equidistant from all three edges, and its radius would be the distance from this line to the edges.But calculating this in 3D is more involved. Let me consider the triangle in the plane x + y + z = 2R with vertices (R, R, 0), (R, 0, R), and (0, R, R).The edges of the triangle are:1. From (R, R, 0) to (R, 0, R): parametric equation (R, R - t, t), t ∈ [0, R]2. From (R, 0, R) to (0, R, R): parametric equation (R - t, t, R), t ∈ [0, R]3. From (0, R, R) to (R, R, 0): parametric equation (t, R, R - t), t ∈ [0, R]The axis of the new cylinder must be a line inside this triangle such that the distance from the axis to each edge is equal to r.To find the maximum r, we need to find the line inside the triangle where the minimum distance to each edge is maximized.This is similar to finding the largest circle that fits inside the triangle, but in 3D, it's a cylinder.In 2D, the largest circle inside a triangle is the incircle, with radius r = (Area)/s, where s is the semiperimeter.In our case, the triangle is equilateral in 3D space, with side length R√2.Wait, the distance between (R, R, 0) and (R, 0, R) is √[(R - R)² + (R - 0)² + (0 - R)²] = √[0 + R² + R²] = √(2R²) = R√2. So, each side is R√2, making it an equilateral triangle with side length R√2.The area of this triangle can be calculated using Heron's formula. The semiperimeter s = (3R√2)/2.Area = √[s(s - a)(s - b)(s - c)] = √[(3R√2/2)(3R√2/2 - R√2)^3] = √[(3R√2/2)(R√2/2)^3] = √[(3R√2/2)(R³√8/8)] = √[(3R√2/2)(R³√8/8)].Wait, this is getting complicated. Alternatively, since it's an equilateral triangle with side length a = R√2, the area is (√3/4)a² = (√3/4)(2R²) = (√3/2)R².The inradius of an equilateral triangle is (a√3)/6 = (R√2 * √3)/6 = R√6/6.But this is in 2D. In 3D, the largest cylinder that fits inside the triangle would have its radius equal to the inradius of the triangle, which is R√6/6.But wait, in 3D, the cylinder's radius is the distance from its axis to each edge of the triangle. So, if the inradius in 2D is R√6/6, then in 3D, the radius of the cylinder would be the same, because the cylinder's axis is along the line equidistant from all edges.Therefore, the radius of the largest cylinder is R√6/6.But let me verify this. The inradius of the equilateral triangle is indeed R√6/6. So, the largest cylinder that fits inside the triangle and touches all three edges has radius R√6/6.But wait, the problem states that the axis of the new cylinder passes inside the triangle, not necessarily along the inradius line. However, to maximize the radius, the axis should be along the line where the distance to all edges is equal, which is the inradius line.Therefore, the radius of the largest cylinder is R√6/6.But let me double-check. The inradius of an equilateral triangle with side length a is (a√3)/6. Here, a = R√2, so inradius is (R√2 * √3)/6 = R√6/6.Yes, that seems correct.So, the radius of the largest cylinder is R√6/6.But wait, in the problem statement, it's mentioned that the axis passes inside the triangle, not necessarily along the inradius. However, to maximize the radius, the axis must be along the inradius line, so the radius is R√6/6.But let me consider another approach. The distance from the centroid to each edge is the inradius. The centroid is at (2R/3, 2R/3, 2R/3). The distance from this point to each edge is R√6/6.Therefore, the radius of the largest cylinder is R√6/6.But wait, in part (a), we found the radius of the sphere to be R(√2 - 1), which is approximately 0.414R, and here, R√6/6 is approximately 0.408R, which is slightly smaller. That seems counterintuitive because the sphere is smaller than the cylinder. Maybe I made a mistake.Wait, no. The sphere is in 3D, while the cylinder is in 3D as well, but the cylinder's radius is measured in a different way. The sphere's radius is the distance from the origin to the surface of the cylinders, while the cylinder's radius is the distance from its axis to the surfaces of the given cylinders.Wait, perhaps I made a mistake in the calculation. Let me recalculate the inradius.For an equilateral triangle with side length a, the inradius is (a√3)/6. Here, a = R√2, so inradius is (R√2 * √3)/6 = R√6/6 ≈ 0.408R.But in part (a), the sphere's radius is R(√2 - 1) ≈ 0.414R, which is slightly larger. That seems contradictory because the sphere is smaller in radius than the cylinder. Wait, no, the sphere is a different shape. The sphere's radius is the distance from the origin to the surface of the cylinders, while the cylinder's radius is the distance from its axis to the surfaces of the given cylinders.Wait, actually, the sphere's radius is R(√2 - 1) ≈ 0.414R, and the cylinder's radius is R√6/6 ≈ 0.408R. So, the sphere is slightly larger in radius than the cylinder. That makes sense because the sphere is a different shape and can fit in a way that the cylinder cannot.Therefore, the radius of the largest cylinder is R√6/6.But let me check another way. The distance from the centroid (2R/3, 2R/3, 2R/3) to each edge of the triangle is R√6/6.Yes, because the distance from a point (x0, y0, z0) to the line defined by two points (x1, y1, z1) and (x2, y2, z2) is |(P2 - P1) × (P1 - P0)| / |P2 - P1|.Let's compute the distance from (2R/3, 2R/3, 2R/3) to the edge from (R, R, 0) to (R, 0, R).Vector P2 - P1 = (R - R, 0 - R, R - 0) = (0, -R, R).Vector P1 - P0 = (R - 2R/3, R - 2R/3, 0 - 2R/3) = (R/3, R/3, -2R/3).Cross product (P2 - P1) × (P1 - P0) = |i j k| |0 -R R| |R/3 R/3 -2R/3|= i[(-R)(-2R/3) - R*(R/3)] - j[(0)(-2R/3) - R*(R/3)] + k[(0)(R/3) - (-R)(R/3)]= i[(2R²/3) - (R²/3)] - j[0 - (R²/3)] + k[0 - (-R²/3)]= i[(R²/3)] - j[ - R²/3 ] + k[ R²/3 ]= (R²/3, R²/3, R²/3)The magnitude of this cross product is √[(R²/3)² + (R²/3)² + (R²/3)²] = √[3*(R^4/9)] = √(R^4/3) = R²/√3.The magnitude of P2 - P1 is √[0² + (-R)² + R²] = √(2R²) = R√2.Therefore, the distance is |(R²/√3)| / (R√2) = (R²/√3) / (R√2) = R / (√3 * √2) = R / √6 = R√6 / 6.Yes, that confirms the inradius is R√6/6.Therefore, the radius of the largest cylinder is R√6/6.But wait, the problem says "the largest cylinder that touches the three given cylinders, if its axis passes inside the triangle with vertices at the points of contact of the three given cylinders."So, the axis passes inside the triangle, which we've considered as the line along the inradius. Therefore, the radius is R√6/6.But let me check if this is the correct answer.Alternatively, considering the symmetry, the radius might be R√2/2, but that seems larger than R√6/6.Wait, R√2/2 ≈ 0.707R, which is larger than R√6/6 ≈ 0.408R. Since the triangle is small, the largest cylinder that fits inside should have a radius smaller than R.But R√6/6 is approximately 0.408R, which seems reasonable.Alternatively, maybe the radius is R(√3 - 1)/√2, but that's approximately 0.517R, which is larger than R√6/6.Wait, perhaps I should consider the distance from the centroid to the edges, which we've calculated as R√6/6, so that should be the radius.Therefore, the radius of the largest cylinder is R√6/6.But let me check another way. The distance from the centroid to each edge is R√6/6, so the radius of the cylinder is R√6/6.Yes, that seems correct.So, the answers are:a) R(√2 - 1)b) R√6/6But let me write them in boxed form.