Math by First Principles
Level 3 · Challenge ← Back to lesson

Composite and Hollow Solids: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered
Question 1 of 12
  1. 1

    A right pipe is 8 cm long, open at both ends, and has wall thickness 2 cm. Its material volume is 192π192\pi cubic centimeters. What is the outer radius?

    A right circular pipe open at both ends. The true end section has outer radius R and inner radius r; the axial section shows an empty bore through the full length 8 cm. Radii are measured from the common center. No computed measurements are shown.A right circular pipe open at both ends. The true end section has outer radius R and inner radius r; the axial section shows an empty bore through the full length 8 cm. Radii are measured from the common center. No computed measurements are shown.RrAxial sectionLength 8 cm
  2. 2

    A solid capsule has radius 2 cm and volume 200π3\frac{200\pi}{3} cubic centimeters. It consists of a right cylinder and two hemispheres of that radius. Find its full tip-to-tip length.

    True axial section of a cylinder with two hemispherical ends. The common radius is 2 cm. The full tip-to-tip length is unknown. Dashed diameters mark internal joins, not exposed disks.True axial section of a cylinder with two hemispherical ends. The common radius is 2 cm. The full tip-to-tip length is unknown. Dashed diameters mark internal joins, not exposed disks.r 2 cm?Tip-to-tip length (schematic)
  3. 3

    A rectangular block has a face 8 cm by 6 cm and unknown perpendicular thickness h. A radius-2 cm cylindrical bore passes through that thickness, strictly inside the face. The retained material volume is (336−28π)(336-28\pi) cubic centimeters. Find h.

    True section perpendicular to a cylindrical through hole in a rectangular block. The outer rectangular face is 8 cm by 6 cm. The centered circular opening has radius 2 cm. The block thickness and bore length is h cm, perpendicular to this face. The hole fits strictly inside the face.True section perpendicular to a cylindrical through hole in a rectangular block. The outer rectangular face is 8 cm by 6 cm. The centered circular opening has radius 2 cm. The block thickness and bore length is h cm, perpendicular to this face. The hole fits strictly inside the face.2 cm8 cm6Thickness h cm
  4. 4

    A side-10 cm cube has a radius-3 cm cylindrical blind cavity drilled 4 cm down from its top, strictly inside that top face. Find all exposed area, including the cavity wall and floor as well as the cube's underside.

    Central vertical section of a side-10 cm cube with a centered radius-3 cm cylindrical blind cavity of depth 4 cm. The cavity stops before reaching the bottom. The axis is dotted; no areas are shown.Central vertical section of a side-10 cm cube with a centered radius-3 cm cylindrical blind cavity of depth 4 cm. The cavity stops before reaching the bottom. The axis is dotted; no areas are shown.r 310Depth 4Cube edge 10 cm
  5. 5

    A right circular cylinder of radius 4 cm has one same-radius hemisphere attached to an end. The exposed boundary consists of the cylinder curved side and hemisphere curved side only; the other flat end is covered by a support. That exposed area is 96π96\pi square centimeters. Find the cylinder length.

    True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 4 cm. The cylinder length is h. Dashed diameters mark internal joins, not exposed disks.True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 4 cm. The cylinder length is h. Dashed diameters mark internal joins, not exposed disks.r 4 cmhCylinder length
  6. 6

    A rectangular box has perpendicular base edges 6 cm and 10 cm. Its space diagonal is 14 cm. What is its exact height?

    A rectangular box with perpendicular edge lengths 6, 10, and h cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.A rectangular box with perpendicular edge lengths 6, 10, and h cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.ABCG6 cm10 cmh
  7. 7

    A right pipe has outer radius 6 cm, length 8 cm, and material volume 160π160\pi cubic centimeters. Both ends are open. Find the area of every exposed material surface, including the inside and both end annuli.

    A right circular pipe open at both ends. The true end section has outer radius 6 cm and inner radius r; the axial section shows an empty bore through the full length 8 cm. Radii are measured from the common center. No computed measurements are shown.A right circular pipe open at both ends. The true end section has outer radius 6 cm and inner radius r; the axial section shows an empty bore through the full length 8 cm. Radii are measured from the common center. No computed measurements are shown.R 6 cmrAxial sectionLength 8 cm
  8. 8

    Two solid parts meet along one matching rectangle 5 cm by 3 cm. Their interiors do not overlap and there is no other contact. The assembled exterior area is 216 square centimeters. What was the sum of their separate total surface areas?

  9. 9

    A right stepped prism has constant depth 3 cm. Its lower block is 10 cm wide and 4 cm high. An upper block of width 4 cm adds unknown height x over the left side. Its total volume is 276 cubic centimeters. Find the added height x.

    Schematic side section of a right stepped prism. Full width 10 cm, lower height 4 cm, upper width 4 cm and added height x cm. Constant perpendicular depth 3 cm. All corners in the section are right angles.Schematic side section of a right stepped prism. Full width 10 cm, lower height 4 cm, upper width 4 cm and added height x cm. Constant perpendicular depth 3 cm. All corners in the section are right angles.10 cm4 cm4xDepth 3 cm
  10. 10

    A rectangular box has perpendicular edges 222\sqrt2 cm, 4 cm, and 7 cm. Find the exact space diagonal.

    A rectangular box with perpendicular edge lengths 2√2, 4, and 7 cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.A rectangular box with perpendicular edge lengths 2√2, 4, and 7 cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.ABCG2√2 cm4 cm7
  11. 11

    A hollow hemispherical bowl is formed by removing a concentric radius-4 cm hemisphere from a radius-6 cm solid hemisphere. The flat circular mouths lie in the same plane. Find all exposed area, including both curved walls and the flat rim annulus.

    True axial section of a hollow hemispherical bowl. Concentric outer and inner radii are 6 cm and 4 cm. Their flat mouths share one plane, leaving an annular rim. The inner central mouth is open; no computed area is shown.True axial section of a hollow hemispherical bowl. Concentric outer and inner radii are 6 cm and 4 cm. Their flat mouths share one plane, leaving an annular rim. The inner central mouth is open; no computed area is shown.6 cm4 cmOOpen mouth plane
  12. 12

    A solid hemisphere of radius 5 cm is joined by its entire flat disk to the base of a right cone of radius 5 cm and perpendicular height 12 cm. Find the exterior area.

    True axial section of a solid hemisphere joined by its full flat disk to a right cone. Common radius 5 cm and cone perpendicular height 12 cm. The dashed diameter is the internal join. No slant length or area is evaluated.True axial section of a solid hemisphere joined by its full flat disk to a right cone. Common radius 5 cm and cone perpendicular height 12 cm. The dashed diameter is the internal join. No slant length or area is evaluated.5 cm12 cm
← Back to lesson