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"Let's consider the x-axis and point P on the x-axis. The cross-section by a plane perpendicular to the x-axis passing through point P forms a right isosceles triangle PQR. Let x be the coordinate of point P, then PQ=QR=√(r^{2}-x^{2)}. Hence, let's denote the area of triangle PQR as S(x), then S(x)=1/2 * PQ * QR = 1/2(r^{2}-x^{2}). Based on this triangle's area, let's find the volume V."

'Consider a rectangular box with side lengths a, b, and c. When the side of length b is used as the axis of rotation and the box is rotated by 90 degrees, the solid formed by all the points the box passes through is denoted as V. (1) Express the volume of V in terms of a, b, and c. (2) When a+b+c=1, find the range of possible values for the volume of V. [University of Tokyo]'

'Cut this solid with a plane perpendicular to the base passing through points Q and S, and further cut with a plane perpendicular to the base passing through points T and R, and color only the faces created by the cuts. Let the solid containing points A, B, C be denoted as solid X, Y, Z respectively. If the volume ratio of solid Y to solid Z is 4:1, answer the following questions.'

'What is the volume of the tetrahedron O-KLMN in comparison to the volume of the rectangular prism ABCD-EFGH?'

'(3) Points P, Q, R are taken on sides AE, BF, CG such that AP:PE=2:1, BQ:QF=1:1, CR:RG=1:2. Figure 2 shows the addition of points P, Q, R to Figure 1. When a solid is formed by cutting pyramid OKLMN with a plane passing through points P, Q, R, what multiple of the volume of the pyramid OKLMN is the volume of the solid that includes point O?'

'(2) If a cube is simultaneously cut by a plane passing through points P, R, and T, and a plane passing through points Q, R, and T, find the ratio of the volumes of the solid formed by point B and the solid formed by point E, in the simplest integer ratio.'

'Answer the following questions. The volume of a cone is calculated as (base area) × (height).\n(1) When assembling this unfolded diagram, which edges touch edge A? Please answer using symbols from edges I to K.\n(2) How many edges does solid C have?\n(3) Let a cube with one edge measuring 6 cm be called D.\nExpress the volume ratio of solid C to cube D in the simplest form of integers.'

'When a plane passing through points P, Q, and F cuts this solid, the plane intersects edge AE at point R.'

'What is the bottom area of water tank A compared to the bottom area of water tank B?'

'Translate the given text into multiple languages.'

'In three-dimensional geometry Z, if the ratio of the area of the colored part to the area of the uncolored part is 1:4, what is the simplest ratio of the surface areas of solid X and solid Z?'

'At this stage, the size cannot be measured because the specimen is out of focus.'

'In a problem of finding the volume of a solid figure, calculate the volumes of the following geometric shapes.'

'When a plane passing through the midpoints of edges AE, BF, CG, DH cuts the pyramid O-KLMN, what is the ratio of the area of the cutting section to the area of the rhombus ABCD?'

'As shown in the diagram, there is a solid with all flat faces, where edge AB is parallel to edge EF, edge BC is parallel to FG, edge CD is parallel to GH, and edge DA is parallel to HE.'

'Let p, q be positive real numbers. Given in the coordinate space with the origin O, three points P(p, 0, 0), Q(0, q, 0), R(0, 0, 1) satisfying ∠PRQ=π/6'

'Given points A(1, -2, -3), B(2, 1, 1), C(-1, -3, 2), D(3, -4, -1). Determine the coordinates of the other vertices of the parallelepiped with line segments AB, AC, and AD as three edges.'

'There is a container in the shape of a hemisphere with a radius of 2 filled with water. When it is tilted gently by an angle α, the water level drops by 197h(2), and the ratio of spilled water to remaining water in the container becomes 11:5. Find the values of h and α. Provide the answer for α in radians.'

'Find the volume V of the solid obtained by rotating the following shape around the x-axis once. (2) The circle x^{2}+(y-2)^{2}=4 and its interior'

'Find the volume of the solid of revolution formed by rotating around the line y=x. Consider the following conditions: the system of inequalities 0 <= x <= t, x² - x <= y <= x, 0 <= t <= 2.'

'When the volume of a sphere increases by 1%, by approximately what percentage do the radius r and the surface area S of the sphere increase?'

'Volume of a tetrahedron'

'Find the volume of the solid formed by rotating the following shape around the x-axis. (1) The area enclosed by the parabola y=-x^{2}+4x and the line y=x.'

'Calculate the volume of the curve represented by the parametric equations and the solid of revolution.'

'In three-dimensional space, let the triangle OAB with vertices O(0,0,0), A(1,0,0), and B(1,1,0) be rotated around the x-axis to form a cone V. Calculate the volume of the solid formed by rotating the cone V around the y-axis.'

'There are two infinitely extending circular cylinders with cross-sections that are circles of radius a. Now, assume that these two cylinders intersect with their central axes forming an angle of π/4. Calculate the volume of the intersection (common part).'

'Let r be a positive real number. In the xyz space, consider the set of points that satisfy the following system of inequalities: x^{2} + y^{2} \\le r^{2}, y^{2} + z^{2} \\ge r^{2}, z^{2} + x^{2} \\le r^{2}. Find the volume of the solid by considering the cross-section by the plane x = t (0 \\le t \\le r).'

'Calculate the volume of a solid of revolution in coordinate space (2).'

'Find the volume of a solid represented by a system of inequalities.'

'Revision'

'Paint each face of a tetrahedron and an octahedron with colors. Each face is painted with only one color. Also, rotating and matching color patterns are considered the same. When there are 12 colors, the number of ways to paint a tetrahedron so that all faces have different colors is A [ ]. Also, when there are 8 colors, the number of ways to paint an octahedron so that all faces have different colors is B [ ].'

'A regular tetrahedron ABCD with side length a is given. (1) Express the radius R of the sphere circumscribed about the regular tetrahedron in terms of a. (2) Express the radius r of the sphere inscribed in the regular tetrahedron in terms of a.'

'How many different ways are there to color the faces of the given polyhedra? Assume that rotationally equivalent colorings are considered the same. (1) Method of coloring each face of a square pyramid with 5 different colors (2) Method of coloring each face of a triangular prism with 5 different colors'

'Basic Example 138 Height and Volume of a Regular Tetrahedron\nLet ABCD be a regular tetrahedron with edge length a.\n(1) Express the height of this tetrahedron in terms of a.\n(2) Express the volume of this tetrahedron in terms of a.'

'There is a cube with a side length of 6 cm. Determine the volume of a regular octahedron formed by the intersection of the diagonals of each face of this cube.'

'From \ \\sqrt{2} \\cos \\theta+1=0 \, we can deduce that \ \\cos \\theta=-\\frac{1}{\\sqrt{2}} \. The point on the semicircle with radius 1, where the x-coordinate is \ -\\frac{1}{\\sqrt{2}} \, is point \ \\mathrm{P} \ in the figure. The angle \ \\theta \ we are looking for is \ \\angle \\mathrm{AOP} \.'

'A cone with height 4 and base radius √2 is tangential to a sphere O on its side and also at the center M of its base. Find the volume V and surface area S of the sphere O.'

'Volume ratio of cube and tetrahedron 102'

"Prove Euler's polyhedron formula: For a convex polyhedron with the number of vertices denoted by v, edges denoted by e, and faces denoted by f, show that v - e + f = 2."

'In the tetrahedron ABCD shown on the right, AD=2, BD=4, CD=6, angles ADB=ADC=BDC=90 degrees, find the following values.'

'In tetrahedron ABCD, where AD=2, BD=4, CD=6, ∠ADB=∠ADC=∠BDC=90°, find the following values:\n(1) Volume V of tetrahedron ABCD\n(2) Area S of △ABC\n(3) Length d of the perpendicular dropped from vertex D to plane ABC'

'A thin metal sheet in the shape of a rectangle where the length is twice the width. From the corners of this sheet, squares with a side length of 1 cm are cut to create an open-top rectangular box. In order to make the volume of the box between 4 cm³ and 24 cm³, what range should the height be?'

'Find the following for the tetrahedron OABC:'

'By extending the height and width of a cube by 2 cm and 1 cm respectively, and reducing the height by 1 cm to form a rectangular prism, the volume increased by 50%. Find the original length of the side of the cube.'

'Find the maximum volume of a right circular cylinder inscribed in a sphere of radius 6. Also, determine the height of the cylinder at that moment.'

'Calculate the volume of a regular tetrahedron'

'Answer the following questions when every vertex of regular icosahedron W with edge length 1 lies on the surface of a sphere S. The icosahedron has all faces congruent equilateral triangles, and each vertex is shared by 5 such triangles.'

'Given a cube with edge length of 1, ABCD-EFGH, a plane containing points A, C, F, and intersecting line BH at points P, and a perpendicular line dropped from point P to the plane ABCD intersecting at point Q.'

'Find the volume of a cylinder and a cone when the base area is S and the height is h. Also, find the volume and surface area of a sphere with radius r.'

'Construct the tetrahedron PABC with the equilateral triangle ABC of side length 3 as the base, where PA=PB=PC=2. Drop a perpendicular PH from vertex P to the base ABC.'

'In tetrahedron ABCD, where AB=3, BC=√13, CA=4, DA=DB=DC=3, drop a perpendicular DH from vertex D to triangle ABC. Find the length of segment DH and the volume of tetrahedron ABCD.'

'Let ABC be an equilateral triangle with side length 3, and PABC be a tetrahedron with PA=PB=PC=2. A perpendicular line PH is dropped from point P to the base ABC.'

'Calculate the height and volume of a regular tetrahedron\nIn a regular tetrahedron ABCD with edge length a, drop a perpendicular line AH from vertex A to triangle BCD.\n(1) Express the length h of AH in terms of a.\n(2) Express the volume V of the regular tetrahedron ABCD in terms of a.\n(3) Express the length of the perpendicular line dropped from point H to triangle ABC in terms of a.'

'Let I be the center of the inscribed sphere. The four tetrahedra IABC, IACD, IABD, IBCD are congruent, so'

'Finding the volume of a regular tetrahedron'

'On the line segment AB of length 6, two points C and D are taken such that AC = BD. Here, 0 < AC < 3. Let S be the sum of the areas of the three circles with the diameters AC, CD, and DB. Find the minimum value of S and the length of the line segment AC at that time.'

'There is a cube A. Reduce A vertically by 1 cm, horizontally by 2 cm, and extend the height by 4 cm to create rectangular prism B. Additionally, extend A vertically by 1 cm, horizontally by 2 cm, and reduce the height by 2 cm to create rectangular prism C. When the volume of A is larger than the volume of B but not larger than the volume of C, find the range of the length of one side of A.'

'Please calculate the volume of the regular icosahedron W.'

'Height and volume of a regular tetrahedron'

'In tetrahedron ABCD, AB = AC = 3, ∠BAC = 90°, AD = 2, BD = CD = √7, and the midpoint of BC is M. In this case, BC = square, DM = triangle, ∠DAM = rectangle degrees, and the volume of tetrahedron ABCD is trapezoid.'

'Find the area of a triangle and its application in solid geometry'

'Find the volume of the tetrahedron ABCD.'

'In the regular tetrahedron ABCD with side length 3, a perpendicular AH is dropped from vertex A to the base BCD. Given that point E on edge AB has AE=1, find:\n(1) sin∠ABH\n(2) The volume of tetrahedron EBCD'

'Find the volume of the tetrahedron OABC. Also, find the distance between point O and the plane ABC.'

'When the volume of a sphere increases by 1%, what percentage does the radius increase by?'

'(2) Let the volume of the container $Q$ be $V_{1}$. For the tetrahedron $ABCD$ with edge length $b$...'

'In the range of 0 ≤ x ≤ π, what is the volume V of the solid formed by rotating the shape enclosed by the curves y=sin x, y=sin 2x around the x-axis?'

'(2) In space, when the center of a sphere with radius 1 moves along one side of a square with side length 4 for one round, find the volume V of the part of the sphere that passes through.'

'In coordinate space, consider a square S with vertices A(-1,1,0), B(1,1,0), C(1,-1,0), and D(-1,-1,0) determined by the inequalities |x| ≤ 1, |y| ≤ 1 in the xy-plane. Let V1 be the solid formed by rotating square S around the line BD and V2 be the solid formed by rotating square S around the line AC.'

'Find the volume V of the solid generated by rotating the region enclosed by the curves and lines around the y-axis once, where y=log(x+1), y=1, and the y-axis.'

'Practice problem 20: Volume of the solid of revolution around the diagonal of a cube'

'Consider a right circular cone with the origin as its center and a circle with the radius as its base, which includes a triangular prism with vertices A(1,1,0), B(1,-1,0), C(-1,-1,0), D(-1,1,0), E(1,0,1), F(-1,0,1). Find the minimum volume of this cone and the radius r of its base at that time.'

'Find the volume V of the solid formed by rotating the region enclosed by the following curves or lines around the y-axis for one full rotation: (1) y=-x^{2}+2, x-axis'

'Let S be a sphere with radius 1 centered at the origin O in the coordinate space. For points A, B, C, D moving on S, let F=2(AB^2+BC^2+CA^2)-3(AD^2+BD^2+CD^2).(1) Let →OA= a, →OB= b, →OC= c, →OD = d, where there exists a constant k different from a, b, c, d, such that F=k( a+ b+ c)·( a+b+c-3d). Find the value of the constant k.(2) Find the maximum value M of F when points A, B, C, D move on S.(3) When the coordinates of point C are (-1/4, √15/4, 0) and the coordinates of point D are (1, 0, 0), determine all pairs of points A and B on S where F=M.'

'When a line segment connecting point P(x, 0) and Q(x, sinx) is used as one side to form an equilateral triangle on a plane perpendicular to the x-axis. As point P moves along the x-axis from the origin O to the point (π, 0), find the volume of the solid formed by this equilateral triangle.'

'Find the radius r of a cone that minimizes the volume of a right circular cone.'

'Tetrahedron ABCD and point P satisfy the equation AP+3BP+2CP+6DP=0.'

'Exercise 3 Maximum volume of a tetrahedron\nIn tetrahedron OABC, where |OA|=a, |OB|=b, |OC|=c, ∠AOB=90°, ∠AOC=α, ∠BOC=β. It is given that 0°<α<90°, 0°<β<90°, and α+β>90°.\n(1) Express the dot products OA⋅OC, OB⋅OC in terms of a, b, c, α, β.\n(2) Let CH be the perpendicular dropped from point C to the plane containing △OAB.\nIf OH=kOA+lOB (where k, l are real numbers), express k, l in terms of a, b, c, α, β.\n(3) Express the volume V of tetrahedron OABC in terms of a, b, c, α, β.\n(4) When a, b, c are constants and α, β satisfy α+β=120°, find the maximum value of V.'

'Example Problem 20 Volume of the solid of rotation around the diagonal of a cube'

'Find the volume V of the solid obtained by rotating the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(x \\geqq 0)$ and the region bounded by the y-axis around the y-axis once. Here, $a>0, b>0$.'

'(3) Find the absolute value of the $x$-coordinate of point A. Points B and P are on the $yz$ plane. Find the height of the tetrahedron POAB and calculate its volume.'

'Consider a right circular cone that is circumscribed about a sphere of radius 1. Find the ratio of the radius and height of the cone with the minimum volume.'

'Find the volume of the solid obtained by rotating the region enclosed by the line y=x and the parabola y=x^2-x around the line OA as the axis, where A is the point of intersection between the line and the parabola (excluding the origin O).'

'For the region bounded by two curves, folding the part below the x-axis about the x-axis results in the gray shaded region in the right diagram. This region is symmetric with respect to the line x=3/4π. Therefore, find the volume V.'

'Volume of Solid of Revolution: Find the volume of the solid obtained by rotating the curve y = √x around the x-axis from x = 0 to x = 1.'

'Find the volume of the solid obtained by rotating the area enclosed by the curve y=-2x^2-1, the x-axis, and the two lines x=-1 and x=2 around the x-axis.'

'A rectangular parallelepiped ABCDEFGH with the lengths of the edges AB=x, AD=y, and AE=z is in space. If the length of the diagonal AG is 3 and the surface area S is 16, then'

'Find the volume of the solid formed by a regular tetrahedron drawn when point P(x, 0) and Q(x, 4-x^2) are connected as one side on a plane perpendicular to the x-axis. As P moves along the x-axis from the origin O to the point (2,0).'

'In a rectangular parallelepiped ABCD-EFGH with AB=x, AD=y, AE=z in space. If the length of the diagonal AG is 3 and the surface area S is 16'

'A rectangular box with no lid was made from a piece of cardboard, measuring 5 cm on each side, by cutting congruent squares from each of its four corners. If the volume of the box is 8 cm³, determine the length of the side of the cut-out squares.'

'A container in the form of an inverted right circular cone with a radius of 5 cm and height of 10 cm. Water is being poured gently into the container at a rate of 2 cm³/s. Determine the following when the depth of water reaches 4 cm:\n(1) The speed at which the water level is rising\n(2) The percentage increase in the water surface area'

'Volume of the solid obtained by rotating S1 around the x-axis once'

'Find the volume V of the following solid of revolution.'

'Point A on the line segment PQ is given by OA = OP + sPQ (0 <= s <= 1)'

'In a three-dimensional space, there are three points O(0,0,0), A(1,0,1), B(0,√3,1). They lie in the plane z=0, and a circle with center O and radius 1 as W. When point P moves along the line segment OA and point Q moves around or inside the circle W, satisfying ⃗OR=⃗OP+⃗OQ, all the points R form a solid V_A. Similarly, when point P moves along the line segment OB and point Q moves around or inside the circle W, satisfying ⃗OR=⃗OP+⃗OQ, all the points R form a solid V_B. Furthermore, let the overlapping part of V_A and V_B be denoted as V. (1) Express the area of the cross-section of the solid V cut by the plane z=cosθ(0≤θ≤π/2) using θ. (2) Determine the volume of the solid V.'

'Find the volume $V$ of the solid obtained by rotating the region enclosed by the curve $x=\\tan \\theta, y=\\cos 2 \\theta$ (for $-\\frac{\\pi}{2}<\\theta<\\frac{\\pi}{2}$) and the $x$-axis around the $x$-axis once.'

'Find the volume of a three-dimensional shape \ V \.'

"There are two infinitely extended cylinders on both sides, with cut sections forming circles of radius a. Now, assume that these cylinders intersect to form a center axis at an angle of π/4. Find the volume of the intersecting part (common part). [Source: Japanese Women's University]"

'Volume of a solid formed by rotating a line in coordinate space'

'Find the volume of the solid obtained by rotating the following shapes around the x-axis: (1) The region enclosed by the parabola y = -x^2 + 4x and the line y = x. (2) The circle x^2 + (y-2)^2 = 4 and its interior.'

'Find the volume V of the solid obtained by rotating the region enclosed by the parabola y=x^{2}-2 x and the line y=-x+2 around the x-axis once.'

'Consider a triangular prism with vertices A(1,1,0), B(1,-1,0), C(-1,-1,0), D(-1,1,0), E(1,0,1), and F(-1,0,1) and a right circular cone with the circle on the xy-plane centered at the origin as the base. Find the minimum volume of such a cone and the radius r of the base at that time.'

'Max and min application problems (2) ... Topic is spatial figures'

'When the line segment connecting points P(x,0) and Q(x, sin x) forms an equilateral triangle with side length 1, and this triangle lies in a plane perpendicular to the x-axis. As point P moves along the x-axis from the origin O to the point (π, 0), find the volume of the solid formed by this equilateral triangle.'

'When the volume V of a sphere increases by 1%, by approximately what percentage do the radius r and the surface area S of the sphere increase?'

'Find the volume V of the solid obtained by rotating the following shapes around the line y=x: (1) The area enclosed by the parabola y=x^2 and the line y=x. (2) The area enclosed by the curve y=sin x (0 <= x <= pi) and the two lines y=x and x+y=pi.'

'Find the volume V of the solid formed by rotating the region enclosed by the following curve or line around the y-axis once.\n(2) y = -x⁴ + 2x² (x ≥ 0), x-axis'

'Challenge question'

'Basic Example 171 Amount of Water Spilled from a Container'

'Curve C: y=x^3 passes through 2 points O(0,0) and A(1,1). Find the volume V of the solid obtained by rotating the region enclosed by curve C and line segment OA around the line OA.'

'Volume of a regular tetrahedron'

'There is a cylinder with a base radius of a and a height of a. When the plane containing the diameter AB of the base and inclined at 30 degrees to the base divides the cylinder into two solids, find the volume V of the smaller solid.'

'In the EX coordinate space, there exists a cylinder that simultaneously satisfies the inequalities $x^{2}+y^{2} \\leqq 1$ and $0 \\leqq z \\leqq 3$. When this cylinder is divided into two solids by a plane passing through the point $(1,0,1)$ and containing the y-axis (forming an angle of $\\frac{\\pi}{4}$ with the 2133 plane), find the volume $V$ of the solid containing the point $(1,0,0)$.'

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'How to calculate volume'

'By decreasing each side of a cube with a side length of 5 cm by 0.02 cm, how much will the surface area and volume of the cube decrease? Please calculate it to two decimal places.'

'In the space with point O(0, 0, 0) as the origin, there are three points A(1, 2, 0), B(0, 2, 3), C(1, 0, 3). Find the volume of tetrahedron OABC.'

'394 Example 62 Vector Equation and Volume Ratio of Tetrahedron'

'Consider the points P(u, u, 0) and Q(u, 0, √(1-u^2)) in 3D space. As u varies from 0 to 1, the surface formed by the line segment PQ is denoted as S.'

'Let the volume of the tetrahedron OABC be V,'

'Find the volume V of the solid obtained by rotating the area enclosed by the curve y = f(x), y-axis, and the line y = f(1) around the y-axis, where f(x)=xe^x+e/2.'

'Solve the math problem: Let y = cos x (0 ≤ x ≤ π/2) represent a curve. The volume of the solid formed by rotating the region enclosed by the curve, the x-axis, and the y-axis around the x-axis is denoted as V1. The volume of a right circular cone with a base radius of 1 and height of π/2 is denoted as V2. Calculate the value of V = V1 - V2.'

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'Solve the maximum and minimum problems related to solid geometry.'

'Find the volume V of the solid generated by rotating the region enclosed by two curves around the x-axis. The curves are defined by y = f(x) and y = g(x) (a <= x <= b, where f(x) >= g(x) >= 0).'

'Find the volume generated by rotating around the y-axis.'

'Find the volume V of the solid obtained by revolving the region bounded by the curve x=f(θ), y=g(θ), the x-axis, and the two lines x=a, x=b (a < b) around the x-axis using the method of cylindrical shells.'

'Find the volume V of the solid obtained by rotating the region enclosed by the curve y=cosx (0 ≤ x ≤ π), y=-1, and the y-axis around the y-axis once.'

'In the space with the origin O as the center, there are three points A(1,2,0), B(0,2,3), C(1,0,3). Find the volume of the tetrahedron OABC.'

'Calculate the volume of the solid obtained by rotating the shape considered in (2) (1) around the y-axis once.'

'Find the volume of the solid obtained by rotating around the y-axis. Explain using the function g(y).'

'Find the volume V of the solid of revolution obtained by rotating the region enclosed by PR curves or lines around the y-axis once.'

'Translate the given question into multiple languages.'

'Find the volume V of the solid obtained by rotating the region enclosed by the curve x=tanθ, y=cos2θ (-π/2<θ<π/2), and the x-axis around the x-axis.'

'Explain how to find the height of a right circular cone.'

'I want to color each face of a regular tetrahedron. However, rotating the tetrahedron to match the coloring is considered the same.\n(1) How many ways are there to color using all four different colors.\n(2) Among the different ways of coloring with three colors, how many ways are there to color using all three colors. Also, if there can be a color among the three that is not used, how many ways are there to color.'

'Example 73 | Volume of a polyhedron\nA polyhedron with 6 squares of side length 3 and 8 equilateral triangles as faces is inscribed in a cube as shown in the figure. Find the volume of this polyhedron. [Setsunan University] Guideline Focus on being inscribed in a cube. Consider a polyhedron formed by cutting off all the corners of the cube by a plane passing through the midpoints of each edge of the cube.'

'A thin metal sheet has a length that is double its width. From the four corners of this sheet, squares with a side length of 5 cm are cut out as shown in the diagram and folded to create an open rectangular container. When the volume of this container is 1.5 L, what are the original dimensions of the sheet in centimeters?'

'Mathematics I'

'There is a tetrahedron PABC with edges of length 3. Find the volume of the remaining shape after removing 4 tetrahedrons of side length 1 from the inside.'