Solid geometry (advanced practice)

{"questions":[{"content":"The area of the lateral surface of a triangular prism is equal to $46.$ A plane parallel to the side edge is drawn through the midsegment of the base of the prism. Find the lateral surface area of the trimmed triangular prism.[[image-1]][[fill_choice_big-40]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d7.svg","width":"200"},"fill_choice_big-40":{"type":"fill_choice_big","options":["$23$","$12$","$15$"],"placeholder":0,"answer":0}},"step":1,"hints":["Note that the cut-off triangular prism has all its lateral sides half the size of the corresponding lateral sides of the original prism, which means that the area of the entire lateral surface will be half as large: $$S = 46:2 = 23$$"]},{"content":"The area of the lateral surface of a triangular prism is equal to $10.$ A plane parallel to the side edge is drawn through the midsegment of the base of the prism. Find the lateral surface area of the trimmed triangular prism.[[image-4]][[fill_choice_big-158]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d7.svg","width":"200"},"fill_choice_big-158":{"type":"fill_choice_big","options":["$5$","$4$","$2$"],"placeholder":0,"answer":0}},"hints":["Note that the cut-off triangular prism has all its lateral sides half the size of the corresponding lateral sides of the original prism, which means that the area of the entire lateral surface will be half as large: $$S = 10:2 = 5$$"]},{"content":"The area of the lateral surface of a triangular prism is equal to $12.$ A plane parallel to the side edge is drawn through the midsegment of the base of the prism. Find the lateral surface area of the trimmed triangular prism.[[image-9]][[fill_choice_big-101]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d7.svg","width":"200"},"fill_choice_big-101":{"type":"fill_choice_big","options":["$6$","$2$","$4$"],"placeholder":0,"answer":0}},"hints":["Note that the cut-off triangular prism has all its lateral sides half the size of the corresponding lateral sides of the original prism, which means that the area of the entire lateral surface will be half as large: $$S = 12:2 = 6$$"]},{"content":"One cylindrical mug is twice as tall as the second, but the second is two and a half times wider. Find the ratio of the volume of the second mug to the volume of the first.[[image-257]][[fill_choice_big-422]]","widgets":{"image-257":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d8.svg","width":"200"},"fill_choice_big-422":{"type":"fill_choice_big","options":["$3.125$","$2.5$","$5$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the cylinder is found by the formula: $$V = \\pi r^2 \\cdot h$$","Change the height and radius according to the problem conditions: $$V = \\pi \\cdot (2.5r)^2 \\cdot \\frac{h}{2} = \\pi \\cdot 6.25r^2 \\cdot \\frac{h}{2}$$ $$V = 3.125\\cdot  \\pi r^2 h$$"]},{"content":"One cylindrical mug is twice as tall as the second, but the second is twice as wide. Find the ratio of the volume of the second mug to the volume of the first.[[image-279]][[fill_choice_big-747]]","widgets":{"image-279":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d8.svg","width":"200"},"fill_choice_big-747":{"type":"fill_choice_big","options":["$2$","$4$","$1$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the cylinder is found by the formula: $$V = \\pi r^2 \\cdot h$$","Change the height and radius according to the problem conditions: $$V = \\pi \\cdot (2r)^2 \\cdot \\frac{h}{2} = \\pi \\cdot 4r^2 \\cdot \\frac{h}{2}$$ $$V = 2 \\cdot  \\pi r^2 h$$"]},{"content":"One cylindrical mug is twice as tall as the second, but the second is three times wider. Find the ratio of the volume of the second mug to the volume of the first.[[image-304]][[fill_choice_big-629]]","widgets":{"image-304":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d8.svg","width":"200"},"fill_choice_big-629":{"type":"fill_choice_big","options":["$4.5$","$1.5$","$6$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the cylinder is found by the formula: $$V = \\pi r^2 \\cdot h$$","Change the height and radius according to the problem conditions: $$V = \\pi \\cdot (3r)^2 \\cdot \\frac{h}{2} = \\pi \\cdot 9r^2 \\cdot \\frac{h}{2}$$ $$V = 4.5\\cdot  \\pi r^2 h$$"]},{"content":"A sphere is circumscribed around the cone (the sphere contains the circumference of the base of the cone and its vertex). The center of the sphere is at the center of the base of the cone. The radius of the sphere is equal to $4\\sqrt{2}.$ Find the slant height of the cone.[[image-984]][[fill_choice_big-1266]]","widgets":{"image-984":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d9.svg","width":"200"},"fill_choice_big-1266":{"type":"fill_choice_big","options":["$8$","$4$","$12$"],"placeholder":0,"answer":0}},"step":1,"hints":["The slant height of the cone can be found using the formula: $$l = \\sqrt{r^2+h^2}$$ Since the cone is inscribed in a sphere, the radius of its base and height are equal to the radius of the sphere: $$l = \\sqrt{r^2+r^2}=\\sqrt{2r^2}$$","$$l = \\sqrt{2\\cdot (4\\sqrt{2})^2}=8$$"]},{"content":"A sphere is circumscribed around the cone (the sphere contains the circumference of the base of the cone and its vertex). The center of the sphere is at the center of the base of the cone. The radius of the sphere is equal to $10\\sqrt{2}.$ Find the slant height of the cone.[[image-1011]][[fill_choice_big-1566]]","widgets":{"image-1011":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d9.svg","width":"198"},"fill_choice_big-1566":{"type":"fill_choice_big","options":["$20$","$10$","$15$"],"placeholder":0,"answer":0}},"step":1,"hints":["The slant height of the cone can be found using the formula: $$l = \\sqrt{r^2+h^2}$$ Since the cone is inscribed in a sphere, the radius of its base and height are equal to the radius of the sphere: $$l = \\sqrt{r^2+r^2}=\\sqrt{2r^2}$$","$$l = \\sqrt{2\\cdot (10\\sqrt{2})^2}=20$$"]},{"content":"A sphere is circumscribed around the cone (the sphere contains the circumference of the base of the cone and its vertex). The center of the sphere is at the center of the base of the cone. The radius of the sphere is equal to $81\\sqrt{2}.$ Find the slant height of the cone.[[image-1040]][[fill_choice_big-1767]]","widgets":{"image-1040":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d9.svg","width":"200"},"fill_choice_big-1767":{"type":"fill_choice_big","options":["$162$","$121$","$100$"],"placeholder":0,"answer":0}},"step":1,"hints":["The slant height of the cone can be found using the formula: $$l = \\sqrt{r^2+h^2}$$ Since the cone is inscribed in a sphere, the radius of its base and height are equal to the radius of the sphere: $$l = \\sqrt{r^2+r^2}=\\sqrt{2r^2}$$","$$l = \\sqrt{2\\cdot (81\\sqrt{2})^2}=162$$"]}]}

{"questions":[{"content":"A cube is circumscribed about a sphere of radius $2.$ Find the volume of the cube.[[image-1]][[fill_choice_big-49]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d4.svg","width":"200"},"fill_choice_big-49":{"type":"fill_choice_big","options":["$64$","$8$","$16$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since the cube is described around a sphere, the length of its side coincides with the diameter of the sphere: $$D = 2r = 2 \\cdot 2 = 4$$","The volume of a cube can be calculated using the formula: $$V = a^3 = 4^3 = 64$$"]},{"content":"A cube is circumscribed about a sphere of radius $3.$ Find the volume of the cube.[[image-4]][[fill_choice_big-122]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d4.svg","width":"200"},"fill_choice_big-122":{"type":"fill_choice_big","options":["$216$","$81$","$27$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since the cube is described around a sphere, the length of its side coincides with the diameter of the sphere: $$D = 2r = 2 \\cdot 3 = 6$$","The volume of a cube can be calculated using the formula: $$V = a^3 = 6^3 = 216$$"]},{"content":"A cube is circumscribed about a sphere of radius $1.$ Find the volume of the cube.[[image-9]][[fill_choice_big-179]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d4.svg","width":"202"},"fill_choice_big-179":{"type":"fill_choice_big","options":["$8$","$1$","$12$"],"placeholder":0,"answer":0}},"hints":["Since the cube is described around a sphere, the length of its side coincides with the diameter of the sphere: $$D = 2r = 2 \\cdot 1 = 2$$","The volume of a cube can be calculated using the formula: $$V = a^3 = 2^3 = 8$$"]},{"content":"The surface area of the ball is $36.$ Find the area of the great circle of the ball.[[image-296]][[fill_choice_big-419]]","widgets":{"image-296":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-419":{"type":"fill_choice_big","options":["$9$","$18$","$6$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a ball is calculated by the formula: $$S = 4 \\pi r^2$$ The area of a circle can be calculated by the formula: $$S = \\pi r^2$$","From the above formulas it is clear that the areas differ by a factor of $4$: $$36:4 = 9$$"]},{"content":"The surface area of the ball is $16.$ Find the area of the great circle of the ball.[[image-311]][[fill_choice_big-596]]","widgets":{"image-311":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-596":{"type":"fill_choice_big","options":["$4$","$8$","$2$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a ball is calculated by the formula: $$S = 4 \\pi r^2$$ The area of a circle can be calculated by the formula: $$S = \\pi r^2$$","From the above formulas it is clear that the areas differ by a factor of $4$: $$16:4 = 4$$"]},{"content":"The surface area of the ball is $40.$ Find the area of the great circle of the ball.[[image-328]][[fill_choice_big-725]]","widgets":{"image-328":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-725":{"type":"fill_choice_big","options":["$10$","$20$","$15$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a ball is calculated by the formula: $$S = 4 \\pi r^2$$ The area of a circle can be calculated by the formula: $$S = \\pi r^2$$","From the above formulas it is clear that the areas differ by a factor of $4$: $$40:4 = 10$$"]},{"content":"Find the volume of a polyhedron whose vertices are the points $A, B, C, A_1, B_1,  C_1$ of a rectangular prism $ABCDA_1B_1C_1D_1,$ whose $AB=10,$ $AD=3,$ $AA_1=9.$[[image-962]][[fill_choice_big-1319]]","widgets":{"image-962":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d6.svg","width":"200"},"fill_choice_big-1319":{"type":"fill_choice_big","options":["$135$","$270$","$80$"],"placeholder":0,"answer":0}},"step":1,"hints":["Note that the polyhedron $A BCA_1 B_1C_1$ occupies exactly half the volume of the original parallelepiped. Find the volume of this parallelepiped and divide by two.","$$V_{cuboid} = abc = 10 \\cdot 3 \\cdot 9 = 270$$ $$V_{A BCA_1 B_1C_1} = 270:2 = 135$$"]},{"content":"Find the volume of a polyhedron whose vertices are the points $A,  B,  C,  A_1,  B_1,  C_1$ of a rectangular prism $ABCDA_1B_1C_1D_1,$ whose $AB=7,$ $AD=6,$ $AA_1=10.$[[image-989]][[fill_choice_big-1635]]","widgets":{"image-989":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d6.svg","width":"200"},"fill_choice_big-1635":{"type":"fill_choice_big","options":["$210$","$420$","$340$"],"placeholder":0,"answer":0}},"step":1,"hints":["Note that the polyhedron $A BCA_1 B_1C_1$ occupies exactly half the volume of the original parallelepiped. Find the volume of this parallelepiped and divide by two.","$$V_{cuboid} = abc = 7 \\cdot 6 \\cdot 10 = 420$$ $$V_{A BCA_1 B_1C_1} = 420:2 = 210$$"]},{"content":"Find the volume of a polyhedron whose vertices are the points $A,  B,  C,  A_1,  B_1,  C_1$ of a rectangular prism $ABCDA_1B_1C_1D_1,$ whose $AB=7,$ $AD=2,$ $AA_1=6.$[[image-1018]][[fill_choice_big-1853]]","widgets":{"image-1018":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d6.svg","width":"202"},"fill_choice_big-1853":{"type":"fill_choice_big","options":["$42$","$84$","$140$"],"placeholder":0,"answer":0}},"step":1,"hints":["Note that the polyhedron $A BCA_1 B_1C_1$ occupies exactly half the volume of the original parallelepiped. Find the volume of this parallelepiped and divide by two.","$$V_{cuboid} = abc = 7 \\cdot 2 \\cdot 6 = 84$$ $$V_{A BCA_1 B_1C_1} = 84:2 = 42$$"]}]}

{"questions":[{"content":"$ 2\\space600\\space cm^3$ of water was poured into a cylindrical vessel. The liquid level turned out to be equal to $10\\space cm.$ The part was completely immersed in water. At the same time, the liquid level in the vessel rose by $2\\space cm.$ Find the volume of the part. Give your answer in $cm^3.$[[image-1]][[fill_choice_big-118]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d1.svg","width":"100"},"fill_choice_big-118":{"type":"fill_choice_big","options":["$520$","$26$","$260$"],"placeholder":0,"answer":0}},"step":1,"hints":["$10\\space cm$ of the vessel holds $ 2\\space600\\space cm^3$ of water, which means $1 \\space cm$ holds: $$2\\space600:10=260$$","Immersion of the part raised the liquid level by $2 \\space cm$: $$260 \\cdot 2 = 520$$"]},{"content":"$ 1\\space500\\space cm^3$ of water was poured into a cylindrical vessel. The liquid level turned out to be equal to $15\\space cm.$ The part was completely immersed in water. At the same time, the liquid level in the vessel rose by $3\\space cm.$ Find the volume of the part. Give your answer in $cm^3.$[[image-5]][[fill_choice_big-191]]","widgets":{"image-5":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d1.svg","width":"100"},"fill_choice_big-191":{"type":"fill_choice_big","options":["$300$","$150$","$900$"],"placeholder":0,"answer":0}},"step":1,"hints":["$15\\space cm$ of the vessel holds $ 1\\space500\\space cm^3$ of water, which means $1 \\space cm$ holds: $$1\\space500:15=100$$","Immersion of the part raised the liquid level by $3 \\space cm$: $$100 \\cdot 3 = 300$$"]},{"content":"$ 180\\space cm^3$ of water was poured into a cylindrical vessel. The liquid level turned out to be equal to $9\\space cm.$ The part was completely immersed in water. At the same time, the liquid level in the vessel rose by $4\\space cm.$ Find the volume of the part. Give your answer in $cm^3.$[[image-12]][[fill_choice_big-268]]","widgets":{"image-12":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d1.svg","width":"100"},"fill_choice_big-268":{"type":"fill_choice_big","options":["$80$","$40$","$10$"],"placeholder":0,"answer":0}},"step":1,"hints":["$9\\space cm$ of the vessel holds $180\\space cm^3$ of water, which means $1 \\space cm$ holds: $$180:9=20$$","Immersion of the part raised the liquid level by $4 \\space cm$: $$20 \\cdot 4 = 80$$"]},{"content":"How many times will the surface area of a cube increase if all its edges are increased by a factor of $15$?[[image-367]][[fill_choice_big-521]]","widgets":{"image-367":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"200"},"fill_choice_big-521":{"type":"fill_choice_big","options":["$225$","$30$","$60$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a cube can be found using the formula: $$S= 6a^2$$","$$S = 6 (15a)^2 = 6a^2 \\cdot 225 $$"]},{"content":"How many times will the surface area of a cube increase if all its edges are increased by a factor of $7$?[[image-396]][[fill_choice_big-728]]","widgets":{"image-396":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"200"},"fill_choice_big-728":{"type":"fill_choice_big","options":["$49$","$42$","$7$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a cube can be found using the formula: $$S= 6a^2$$","$$S = 6 (7a)^2 = 6 a^2 \\cdot 49$$"]},{"content":"How many times will the surface area of a cube increase if all its edges are increased by a factor of $10$?[[image-421]][[fill_choice_big-883]]","widgets":{"image-421":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"200"},"image-846":{"type":"image","url":""},"fill_choice_big-883":{"type":"fill_choice_big","options":["$100$","$60$","$1000$"],"placeholder":0,"answer":0}},"step":1,"hints":["The surface area of a cube can be found using the formula: $$S= 6a^2$$","$$S = 6 (10a)^2 = 6 a^2 \\cdot 100$$"]},{"content":"The volume of the first cube is $64$ times greater than the volume of the second cube. How many times is the surface area of the first cube greater than the surface area of the second cube?[[image-1205]][[fill_choice_big-1554]]","widgets":{"image-1205":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d3.svg","width":"201"},"fill_choice_big-1554":{"type":"fill_choice_big","options":["$16$","$32$","$8$"],"placeholder":0,"answer":0},"fill_choice_big-1857":{"type":"fill_choice_big","options":["$16$","$32$","$8$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a cube is calculated by the formula: $$V = a^3$$ The volumes of cubes differ by a factor of $8$, which means their sides are in the ratio: $$\\sqrt[3]{64}= 4$$","The surface area of a cube is found by the formula: $$S = 6 a^2$$ Since the sides differ by a factor of two, the areas will be related as: $$4^2 =16$$"]},{"content":"The volume of the first cube is $8$ times greater than the volume of the second cube. How many times is the surface area of the first cube greater than the surface area of the second cube?[[image-1234]][[fill_choice_big-2106]]","widgets":{"image-1234":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d3.svg","width":"201"},"fill_choice_big-2106":{"type":"fill_choice_big","options":["$4$","$2$","$16$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a cube is calculated by the formula: $$V = a^3$$ The volumes of cubes differ by a factor of $8$, which means their sides are in the ratio: $$\\sqrt[3]{8}= 2$$","The surface area of a cube is found by the formula: $$S = 6 a^2$$ Since the sides differ by a factor of two, the areas will be related as: $$2^2 =4$$"]},{"content":"The volume of the first cube is $27$ times greater than the volume of the second cube. How many times is the surface area of the first cube greater than the surface area of the second cube?[[image-1265]][[fill_choice_big-2350]]","widgets":{"image-1265":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/d3.svg","width":"201"},"fill_choice_big-2350":{"type":"fill_choice_big","options":["$9$","$27$","$54$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a cube is calculated by the formula: $$V = a^3$$ The volumes of cubes differ by a factor of $8$, which means their sides are in the ratio: $$\\sqrt[3]{27}= 3$$","The surface area of a cube is found by the formula: $$S = 6 a^2$$ Since the sides differ by a factor of two, the areas will be related as: $$3^2 =9$$"]}]}

{"questions":[{"content":"Find the volume of a polyhedron whose vertices are the points $A, A_1, B_1, C_1$ of a regular triangular prism $ABCA_1B_1C_1,$ whose base area is equal to $10,$ and whose lateral edge is equal to $3.$[[image-1]][[fill_choice_big-60]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k8.svg","width":"200"},"fill_choice_big-60":{"type":"fill_choice_big","options":["$10$","$15$","$20$"],"placeholder":0,"answer":0}},"step":1,"hints":["The figure $A A_1B_1 C_1$ is a pyramid. The volume of the pyramid is calculated by the formula: $$V = \\frac{1}{3}S_{base} \\cdot h$$","$$V = \\frac{1}{3}\\cdot 10 \\cdot 3=10$$"]},{"content":"Find the volume of a polyhedron whose vertices are the points $A, A_1, B_1, C_1$ of a regular triangular prism $ABCA_1B_1C_1,$ whose base area is equal to $3,$ and whose lateral edge is equal to $8.$[[image-6]][[fill_choice_big-117]]","widgets":{"image-6":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k8.svg","width":"200"},"fill_choice_big-117":{"type":"fill_choice_big","options":["$8$","$24$","$16$"],"answer":0}},"step":1,"hints":["The figure $A A_1B_1 C_1$ is a pyramid. The volume of the pyramid is calculated by the formula: $$V = \\frac{1}{3}S_{base} \\cdot h$$","$$V = \\frac{1}{3}\\cdot 3\\cdot 8=8$$"]},{"content":"Find the volume of a polyhedron whose vertices are the points $A, A_1, B_1, C_1$ of a regular triangular prism $ABCA_1B_1C_1,$ whose base area is equal to $10,$ and whose lateral edge is equal to $9.$[[image-11]][[fill_choice_big-180]]","widgets":{"image-11":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k8.svg","width":"200"},"fill_choice_big-180":{"type":"fill_choice_big","options":["$30$","$90$","$45$"],"placeholder":0,"answer":0}},"step":1,"hints":["The figure $A A_1B_1 C_1$ is a pyramid. The volume of the pyramid is calculated by the formula: $$V = \\frac{1}{3}S_{base} \\cdot h$$","$$V = \\frac{1}{3}\\cdot 10\\cdot 9=30$$"]},{"content":"A sphere whose volume is equal to $46,$ is inscribed in a cylinder. Find the volume of the cylinder.[[image-273]][[fill_choice_big-495]]","widgets":{"image-273":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k10.svg","width":"200"},"fill_choice_big-495":{"type":"fill_choice_big","options":["$69$","$23$","$92$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since the ball is inscribed in a cylinder, the radius of the ball is equal to the radius of the base of the cylinder, and the diameter of the ball is equal to the height of the cylinder. The volume of the ball is calculated by the formula: $$V_{sphere} = \\frac{4}{3} \\pi r^3$$ The volume of the cylinder is calculated by the formula: $$V_{cylinder} = \\pi r^2 \\cdot h$$ But since the height of the cylinder is the diameter of the ball, we can write: $$V_{cylinder} = \\pi r^2 \\cdot 2r=2 \\pi r^3$$","Thus, the volumes of the figures differ by a factor of $\\frac{2}{3}$: $$46:\\frac{2}{3} = 69$$"]},{"content":"A sphere whose volume is equal to $8,$ is inscribed in a cylinder. Find the volume of the cylinder.[[image-288]][[fill_choice_big-673]]","widgets":{"image-288":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k10.svg","width":"200"},"fill_choice_big-673":{"type":"fill_choice_big","options":["$12$","$16$","$14$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since the ball is inscribed in a cylinder, the radius of the ball is equal to the radius of the base of the cylinder, and the diameter of the ball is equal to the height of the cylinder. The volume of the ball is calculated by the formula: $$V_{sphere} = \\frac{4}{3} \\pi r^3$$ The volume of the cylinder is calculated by the formula: $$V_{cylinder} = \\pi r^2 \\cdot h$$ But since the height of the cylinder is the diameter of the ball, we can write: $$V_{cylinder} = \\pi r^2 \\cdot 2r=2 \\pi r^3$$","Thus, the volumes of the figures differ by a factor of $\\frac{2}{3}$: $$8:\\frac{2}{3} = 12$$"]},{"content":"A sphere whose volume is equal to $10,$ is inscribed in a cylinder. Find the volume of the cylinder.[[image-305]][[fill_choice_big-780]]","widgets":{"image-305":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k10.svg","width":"200"},"fill_choice_big-780":{"type":"fill_choice_big","options":["$15$","$20$","$25$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since the ball is inscribed in a cylinder, the radius of the ball is equal to the radius of the base of the cylinder, and the diameter of the ball is equal to the height of the cylinder. The volume of the ball is calculated by the formula: $$V_{sphere} = \\frac{4}{3} \\pi r^3$$ The volume of the cylinder is calculated by the formula: $$V_{cylinder} = \\pi r^2 \\cdot h$$ But since the height of the cylinder is the diameter of the ball, we can write: $$V_{cylinder} = \\pi r^2 \\cdot 2r=2 \\pi r^3$$","Thus, the volumes of the figures differ by a factor of $\\frac{2}{3}$: $$10:\\frac{2}{3} = 15$$"]},{"content":"The volume of the cone is equal to $88.$ Through the middle of the height, parallel to the base of the cone, a section is drawn, which is the base of a smaller cone with the same vertex. Find the volume of the smaller cone.[[image-1016]][[fill_choice_big-1223]]","widgets":{"image-1016":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k12.svg","width":"200"},"fill_choice_big-1223":{"type":"fill_choice_big","options":["$11$","$44$","$33$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volumes of similar figures are related as the cube of the scale factor. The scale factor of the resulting cones is equal to two, which means that the volume of the smaller cone is: $$88:2^3 = 11$$"]},{"content":"The volume of the cone is equal to $64.$ Through the middle of the height, parallel to the base of the cone, a section is drawn, which is the base of a smaller cone with the same vertex. Find the volume of the smaller cone.[[image-1043]][[fill_choice_big-1492]]","widgets":{"image-1043":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k12.svg","width":"200"},"fill_choice_big-1492":{"type":"fill_choice_big","options":["$8$","$32$","$21$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volumes of similar figures are related as the cube of the scale factor. The scale factor of the resulting cones is equal to two, which means that the volume of the smaller cone is: $$64:2^3 = 8$$"]},{"content":"The volume of the cone is equal to $8.$ Through the middle of the height, parallel to the base of the cone, a section is drawn, which is the base of a smaller cone with the same vertex. Find the volume of the smaller cone.[[image-1072]][[fill_choice_big-1709]]","widgets":{"image-1072":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k12.svg","width":"200"},"fill_choice_big-1709":{"type":"fill_choice_big","options":["$1$","$4$","$3$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volumes of similar figures are related as the cube of the scale factor. The scale factor of the resulting cones is equal to two, which means that the volume of the smaller cone is: $$8:2^3 = 1$$"]}]}

{"questions":[{"content":"At the base of a straight prism lies a right triangle with legs $7$ and $5.$. The lateral edges of the prism are equal to $\\frac{8}{\\pi}$. Find the volume of the cylinder circumscribed around this prism.[[image-1]][[fill_choice_big-94]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k5.svg","width":"200"},"fill_choice_big-94":{"type":"fill_choice_big","options":["$148$","$74$","$35$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since at the base of the prism a right triangle is inscribed in a circle, its hypotenuse will be the diameter of this circle: $$D=\\sqrt{a^2+b^2}=\\sqrt{7^2+5^2}=\\sqrt{74}$$ The radius is equal to half the diameter: $$R = \\frac{D}{2} = \\frac{\\sqrt{74}}{2}$$","The volume of the cylinder is equal to the product of the base and the height, which coincides with the height of the prism: $$V=S_{base} \\cdot h$$ $$V=\\pi R^2 \\cdot h$$ $$V = \\pi \\cdot \\Big({ \\frac{\\sqrt{74}}{2}}\\Big)^2 \\cdot \\frac{8}{\\pi}=148$$"]},{"content":"At the base of a straight prism lies a right triangle with legs $1$ and $5.$. The lateral edges of the prism are equal to $\\frac{4}{\\pi}$. Find the volume of the cylinder circumscribed around this prism.[[image-4]][[fill_choice_big-171]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k5.svg","width":"200"},"fill_choice_big-171":{"type":"fill_choice_big","options":["$26$","$52$","$20$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since at the base of the prism a right triangle is inscribed in a circle, its hypotenuse will be the diameter of this circle: $$D=\\sqrt{a^2+b^2}=\\sqrt{1^2+5^2}=\\sqrt{26}$$ The radius is equal to half the diameter: $$R = \\frac{D}{2} = \\frac{\\sqrt{26}}{2}$$","The volume of the cylinder is equal to the product of the base and the height, which coincides with the height of the prism: $$V=S_{base} \\cdot h$$ $$V=\\pi R^2 \\cdot h$$ $$V = \\pi \\cdot \\Big({ \\frac{\\sqrt{26}}{2}}\\Big)^2 \\cdot \\frac{4}{\\pi}=26$$"]},{"content":"At the base of a straight prism lies a right triangle with legs $8$ and $1.$. The lateral edges of the prism are equal to $\\frac{4}{\\pi}$. Find the volume of the cylinder circumscribed around this prism.[[image-9]][[fill_choice_big-233]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k5.svg","width":"200"},"fill_choice_big-233":{"type":"fill_choice_big","options":["$65$","$130$","$64$"],"placeholder":0,"answer":0}},"step":1,"hints":["Since at the base of the prism a right triangle is inscribed in a circle, its hypotenuse will be the diameter of this circle: $$D=\\sqrt{a^2+b^2}=\\sqrt{8^2+1^2}=\\sqrt{65}$$ The radius is equal to half the diameter: $$R = \\frac{D}{2} = \\frac{\\sqrt{65}}{2}$$","The volume of the cylinder is equal to the product of the base and the height, which coincides with the height of the prism: $$V=S_{base} \\cdot h$$ $$V=\\pi R^2 \\cdot h$$ $$V = \\pi \\cdot \\Big({ \\frac{\\sqrt{65}}{2}}\\Big)^2 \\cdot \\frac{4}{\\pi}=65$$"]},{"content":"The area of the great circle of the ball is $23.$ Find the surface area of the ball.[[image-337]][[fill_choice_big-488]]","widgets":{"image-337":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-488":{"type":"fill_choice_big","options":["$92$","$46$","$69$"],"placeholder":0,"answer":0}},"step":1,"hints":["The area of a circle is calculated by the formula: $$S_{circle} = \\pi r^2$$ The surface area of a ball is calculated by the formula: $$S_{sphere}= 4 \\pi r^2$$","Find the surface area of the ball: $$ 4 \\cdot 23 = 92$$"]},{"content":"The area of the great circle of the ball is $15.$ Find the surface area of the ball.[[image-373]][[fill_choice_big-782]]","widgets":{"image-373":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-782":{"type":"fill_choice_big","options":["$60$","$30$","$20$"],"placeholder":0,"answer":0}},"step":1,"hints":["The area of a circle is calculated by the formula: $$S_{circle} = \\pi r^2$$ The surface area of a ball is calculated by the formula: $$S_{sphere}= 4 \\pi r^2$$","Find the surface area of the ball: $$ 4 \\cdot 15 = 60$$"]},{"content":"The area of the great circle of the ball is $101.$ Find the surface area of the ball.[[image-406]][[fill_choice_big-675]]","widgets":{"image-406":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k6.svg","width":"200"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$404$","$101$","$202$"],"placeholder":0,"answer":0}},"hints":["The area of a circle is calculated by the formula: $$S_{circle} = \\pi r^2$$ The surface area of a ball is calculated by the formula: $$S_{sphere}= 4 \\pi r^2$$","Find the surface area of the ball: $$ 4 \\cdot 101 = 404$$"]},{"content":"The volume of the parallelepiped $ABCDA_1B_1C_1D_1$ is equal to $36.$ Find the volume of the triangular pyramid $ABDA_1.$[[image-983]][[fill_choice_big-1319]]","widgets":{"image-983":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k7.svg","width":"300"},"fill_choice_big-1319":{"type":"fill_choice_big","options":["$6$","$12$","$18$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the pyramid is equal to $\\frac{1}{3}$ the volume of the parallelepiped, the base and height of which coincide, but since the base of the pyramid is half the size of the base of the parallelepiped, the volume must be reduced by another factor: $$V = \\frac{1}{3} \\cdot \\frac{1}{2} \\cdot 36 = 6$$"]},{"content":"The volume of the parallelepiped $ABCDA_1B_1C_1D_1$ is equal to $54.$ Find the volume of the triangular pyramid $ABDA_1.$[[image-1036]][[fill_choice_big-1556]]","widgets":{"image-1036":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k7.svg","width":"300"},"fill_choice_big-1556":{"type":"fill_choice_big","options":["$9$","$6$","$27$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the pyramid is equal to $\\frac{1}{3}$ the volume of the parallelepiped, the base and height of which coincide, but since the base of the pyramid is half the size of the base of the parallelepiped, the volume must be reduced by another factor: $$V = \\frac{1}{3} \\cdot \\frac{1}{2} \\cdot 54 = 9$$"]},{"content":"The volume of the parallelepiped $ABCDA_1B_1C_1D_1$ is equal to $60.$ Find the volume of the triangular pyramid $ABDA_1.$[[image-1093]][[fill_choice_big-1791]]","widgets":{"image-1093":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k7.svg","width":"300"},"fill_choice_big-1791":{"type":"fill_choice_big","options":["$10$","$30$","$20$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of the pyramid is equal to $\\frac{1}{3}$ the volume of the parallelepiped, the base and height of which coincide, but since the base of the pyramid is half the size of the base of the parallelepiped, the volume must be reduced by another factor: $$V = \\frac{1}{3} \\cdot \\frac{1}{2} \\cdot 60 = 10$$"]}]}

{"questions":[{"content":"In a cylindrical vessel, the liquid level reaches $18\\space cm.$ At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of which is $3$ times greater than the diameter of the first? Give your answer in centimeters.[[image-1]][[fill_choice_big-61]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k1.svg","width":"300"},"fill_choice_big-61":{"type":"fill_choice_big","options":["$2$","$9$","$6$"],"placeholder":0,"answer":0}},"step":1,"hints":["The area of the bottom of the second vessel will be $3^2$ times larger, since the area of the circle depends on the square of the radius: $$S=\\pi r^2$$","This means that the height of the liquid level will be divided by $9$: $$18:9=2$$"]},{"content":"In a cylindrical vessel, the liquid level reaches $8\\space cm.$ At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of which is $2$ times greater than the diameter of the first? Give your answer in centimeters.[[image-6]][[fill_choice_big-36]]","widgets":{"image-6":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k1.svg","width":"297"},"fill_choice_big-36":{"type":"fill_choice_big","options":["$2$","$4$","$1$"],"placeholder":0,"answer":0}},"step":1,"hints":["The area of the bottom of the second vessel will be $2^2$ times larger, since the area of the circle depends on the square of the radius: $$S=\\pi r^2$$","This means that the height of the liquid level will be divided by $4$: $$8:4=2$$"]},{"content":"In a cylindrical vessel, the liquid level reaches $336\\space cm.$ At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of which is $4$ times greater than the diameter of the first? Give your answer in centimeters.[[image-17]][[fill_choice_big-92]]","widgets":{"image-17":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k1.svg","width":"300"},"fill_choice_big-92":{"type":"fill_choice_big","options":["$21$","$168$","$16$"],"placeholder":0,"answer":0}},"step":1,"hints":["The area of the bottom of the second vessel will be $4^2$ times larger, since the area of the circle depends on the square of the radius: $$S=\\pi r^2$$","This means that the height of the liquid level will be divided by $16$: $$336:16=21$$"]},{"content":"How many times will the volume of a cube increase if all its edges are increased five times?[[image-347]][[fill_choice_big-484]]","widgets":{"image-347":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"300"},"fill_choice_big-484":{"type":"fill_choice_big","options":["$125$","$5$","$25$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a cube is found by the formula: $$V=a^3$$","If the edge of the cube is increased by $5$ times, we get: $$V = (5a)^3 = 125a^3$$"]},{"content":"How many times will the volume of a cube increase if all its edges are doubled?[[image-376]][[fill_choice_big-465]]","widgets":{"image-376":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"300"},"fill_choice_big-465":{"type":"fill_choice_big","options":["$8$","$4$","$2$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a cube is found by the formula: $$V=a^3$$","If the edge of the cube is increased by $2$ times, we get: $$V = (2a)^3 = 8a^3$$"]},{"content":"How many times will the volume of a cube increase if all its edges are tripled?[[image-409]][[fill_choice_big-446]]","widgets":{"image-409":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k2-3.svg","width":"300"},"fill_choice_big-446":{"type":"fill_choice_big","options":["$27$","$9$","$3$"],"placeholder":0,"answer":0}},"hints":["The volume of a cube is found by the formula: $$V=a^3$$","If the edge of the cube is increased by $3$ times, we get: $$V = (3a)^3 = 27a^3$$"]},{"content":"The cylinder is inscribed in a rectangular prism. The radius of the base and the height of the cylinder are equal to $15.$ Find the volume of the parallelepiped.[[image-1070]][[fill_choice_big-1247]]","widgets":{"image-1070":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k4.svg","width":"300"},"fill_choice_big-1247":{"type":"fill_choice_big","options":["$13 \\space 500$","$225$","$3\\space375$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a parallelepiped can be found by multiplying its length, width and height: $$V=abc$$","Since the cylinder is inscribed in a parallelepiped, its diameter coincides with the width and length of the parallelepiped. Diameter is twice the radius:$$D=2r = 2 \\cdot 15 = 30 $$","Since the cylinder is inscribed in a parallelepiped, its height coincides with the height of the parallelepiped. Find the volume of the parallelepiped: $$V = 30 \\cdot 30 \\cdot 15 = 13 \\space 500$$"]},{"content":"The cylinder is inscribed in a rectangular prism. The radius of the base and the height of the cylinder are equal to $3.$ Find the volume of the parallelepiped.[[image-1097]][[fill_choice_big-1356]]","widgets":{"image-1097":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k4.svg","width":"300"},"fill_choice_big-1356":{"type":"fill_choice_big","options":["$108$","$27$","$9$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a parallelepiped can be found by multiplying its length, width and height: $$V=abc$$","Since the cylinder is inscribed in a parallelepiped, its diameter coincides with the width and length of the parallelepiped. Diameter is twice the radius:$$D=2r = 2 \\cdot 3 = 6 $$","Since the cylinder is inscribed in a parallelepiped, its height coincides with the height of the parallelepiped. Find the volume of the parallelepiped: $$V = 6 \\cdot 6 \\cdot 3 = 108$$"]},{"content":"The cylinder is inscribed in a rectangular prism. The radius of the base and the height of the cylinder are equal to $2.$ Find the volume of the parallelepiped.[[image-1126]][[fill_choice_big-1472]]","widgets":{"image-1126":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/12/k4.svg","width":"300"},"fill_choice_big-1472":{"type":"fill_choice_big","options":["$32$","$16$","$64$"],"placeholder":0,"answer":0}},"step":1,"hints":["The volume of a parallelepiped can be found by multiplying its length, width and height: $$V=abc$$","Since the cylinder is inscribed in a parallelepiped, its diameter coincides with the width and length of the parallelepiped. Diameter is twice the radius:$$D=2r = 2 \\cdot 2 = 4 $$","Since the cylinder is inscribed in a parallelepiped, its height coincides with the height of the parallelepiped. Find the volume of the parallelepiped: $$V = 4 \\cdot 4 \\cdot 2 = 32$$"]}]}