Frustum practice problems
A frustum (plural: frusta or frustums) is a geometric shape that results from cutting the top off a cone or pyramid with a plane parallel to its base. It is essentially a truncated cone or pyramid, consisting of two parallel bases (a larger base and a smaller top) connected by a curved or slanted surface.Key Features of a Frustum:
1. Bases: Two parallel polygonal or circular faces (one larger and one smaller).
2. Height (h): The perpendicular distance between the two bases.
3. Slant Height (l): The distance along the lateral (side) surface between the edges of the two bases (applies to cones and pyramids).
Types of Frustums:
1. Frustum of a Cone:
- The two bases are circles.
- The lateral surface is a curved surface.
2. Frustum of a Pyramid:
- The two bases are polygons (e.g., square, triangle).
- The lateral surface consists of trapezoidal faces.
Directions: Provide a careful solution to each problem, showing all steps in your work.
Number of problems found: 57
- A cone 4
A cone with a radius of 10 cm is divided into two parts by drawing a plane through the midpoint of its axis parallel to its base. Compare the volumes of the two parts. - A lamp
A lamp shade like that of a frustum has a height of 12 cm and an upper and lower diameter of 10 cm and 20 cm. What area of materials is required to cover the curved surface of the frustum? - Lamp shade
A lampshade is in the form of a frustum of a cone with a slant height of 7 inches and base radii of 3 inches and 7 inches respectively. How much material is used in its construction if 1/4 inch is allowed for the seam? - A frustum
A frustum of a pyramid consists of a square base of length 10 cm and a top square of length 7 cm. The height of the frustum is 6 cm. Calculate the surface area and volume. - The frustum A frustum of a pyramid is 4 cm at the top and 7 cm at the bottom square, and it's 6 cm high. Calculate the volume of the frustum.
- Slant height 3
The frustum of a right circular cone has the diameters of base 10 cm of top 6 cm and a height of 5 cm. Find the slant height. - Side deviation
A frustum has base radii r₁ and r₂ where r₁ > r₂, r₂ = s, and the slant side makes an angle of 60° with the base plane. Express the surface area and volume of the frustum in terms of its slant height s. - Wooden bowls
Twenty wooden bowls in the shape of a truncated cone should be painted on the outside and inside with wood varnish. We need 0.1 l of paint to paint 200 cm². How many liters of paint do we have to buy if the bowls are 25 cm high, the bottom of the bowl has - Hourglass
An hourglass consists of two identical containers in the shape of rotational cones. For simplicity, we assume that the cones touch only at their apexes. The sand reaches to half the height of the lower cone. After turning the hourglass over, it takes exac - Runcated pyramid teapot
The 35 cm high teapot has the shape of a truncated pyramid with the length of the edge of the lower square base a=50 cm and the edges of the rectangular base b: 20 cm and c: 30 cm. How many liters of water will fit in the teapot? - 2x cone
A circular cone of height 36 cm is cut by a plane parallel to the base, dividing it into two smaller cones of equal volume. Calculate the height of the smaller cone. - Pit
The pit is 1.2 m deep and in the shape of a truncated pyramid with a rectangular base. Its length and width are the top 3 × 1.5 m and the bottom 1 m × 0.5 m. To paint one square meter of the pit, we use 0.8 l of green paint. How many liters of paint are n - Frustum of a cone
A reservoir contains 28.54 m³ of water when complete. The diameter of the upper base is 3.5 m, while the lower base is 2.5 m. Find the height if the reservoir is in the form of a frustum of a right circular cone. - Truncated cone
Calculate the height of the rotating truncated cone with volume V = 1471 cm³ and a base radii r1 = 6.1 cm and r2 = 7.9 cm. - Tetrahedron water level
A container shaped like a rotating cylinder with a base radius of 5 cm is filled with water. If a regular tetrahedron with an edge of 7 cm is immersed in it, how much will the water level in the container rise? - Cone - bases
The volume of the cut cone is V = 38000π cm³. The radius of the lower base is 10 cm larger than the radius of the upper base. Determine the radius of the base if height v = 60 cm. - The surface
The surface of a truncated rotating cone with side s = 13 cm is S = 510π cm². Find the radii of the bases when their difference in lengths is 10 cm. - Flowerbed
The flowerbed has the shape of a truncated pyramid. The bottom edge of the base a = 10 m, and the upper base b = 9 m. The deviation angle between the edge and the base is alpha = 45°. What volume is needed to make this flowerbed? How many plants can be pl - Similar frustums
The upper and lower radii of a frustum of a right circular cone are 8 cm and 32 cm, respectively. If the altitude of the frustum is 10 cm, how far from the bottom base must a cutting plane be made to form two similar frustums? - Hexagon cut pyramid
Calculate the volume of a regular 6-sided cut pyramid if the bottom edge is 30 cm, the top edge is 12 cm, and the side edge length is 41 cm.
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