Tenth Grade Mathematics: Composite Solids: Toy, Capsule, and Boiler Problems
In mechanical engineering, manufacturing, and consumer product design, industrial artifacts are constructed by joining multiple basic geometric shapes into composite solids. A medicine capsule is composed of a cylinder capped with two hemispherical ends. A spinning toy is constructed from a cone mounted on a hemisphere. An industrial storage boiler consists of a cylindrical tank with conical or hemispherical headers. Analyzing these composite bodies requires a strict distinction between volume (which is directly additive) and surface area (where joining interfaces vanish inside the interior).
The Principle of Non-Overlapping Volume
Volume is an additive physical measure: the total space occupied by a composite solid is the simple arithmetic sum of the volumes of its constituent components:
The Composite Volume Axiom:
For any solid formed by joining distinct geometric parts A, B, and C:
Total Volume = Volume(A) + Volume(B) + Volume(C)
No interior adjustments are necessary when calculating volume because the interior cavities combine without reduction.
The Critical Rule of Composite Surface Area
The most widespread misconception among mathematics students is assuming that surface areas can be added together directly. When two solid faces are glued or welded together, those faces cease to be part of the exterior boundary!
THE GOLDEN RULE OF COMPOSITE SURFACE AREA:
Only the VISIBLE, EXTERNAL surfaces that are exposed to the surrounding air
contribute to the Total Surface Area of the composite solid!
The joining interfaces that touch inside the solid MUST BE EXCLUDED!
Visualizing Vanishing Interfaces:
Conical Cap: / / \ Curved Surface exposed to air
+----+ Flat circular base (JOINT INTERFACE - VANISHES!)
|====|
Cylindrical Body: | | Lateral wall exposed to air
| |
+----+ Flat circular base (JOINT INTERFACE - VANISHES!)
|====|
Hemispherical Base: \ / Curved hemispherical surface exposed to air
\__/
TSA of Composite Toy = CSA(Cone) + CSA(Cylinder) + CSA(Hemisphere)
(None of the internal circular bases are included!)
Category 1: The Spinning Toy (Cone Mounted on Hemisphere)
A spinning top (or toy) is constructed by mounting a right circular cone of radius r on the flat circular top of a hemisphere of identical radius r:
Formulas for Toy:
Let cone have base radius r and height h.
Let hemisphere have radius r.
1. Slant Height of Cone: l = sqrt(r^2 + h^2)
2. Total Surface Area of Toy = CSA of Cone + CSA of Hemisphere
= pi * r * l + 2 * pi * r^2
= pi * r * (l + 2*r)
3. Total Volume of Toy = Volume of Cone + Volume of Hemisphere
= (1/3) * pi * r^2 * h + (2/3) * pi * r^3
= (1/3) * pi * r^2 * (h + 2*r)
Category 2: The Medicine Capsule (Cylinder with Two Hemispherical Ends)
A capsule is shaped like a cylinder with two hemispheres stuck to each of its ends:
Parameters of Capsule:
Total Length of Capsule = L
Diameter of Capsule = d ===> Radius r = d / 2
The two hemispherical ends each have radius r.
Length of Cylindrical Portion: h = L - 2*r
Formulas:
1. Total Surface Area = CSA(Cylinder) + 2 * CSA(Hemisphere)
= 2*pi*r*h + 2*(2*pi*r^2) = 2*pi*r*h + 4*pi*r^2
= 2 * pi * r * (h + 2*r)
Notice: (h + 2*r) = L (Total Length!)
TSA = 2 * pi * r * L = pi * d * L
2. Total Volume = Volume(Cylinder) + 2 * Volume(Hemisphere)
= pi * r^2 * h + 2 * [(2/3) * pi * r^3]
= pi * r^2 * [h + (4/3)*r]
Category 3: Cubical Block with Hemispherical Surmounting or Depression
A solid cubical block of edge a has a hemisphere of diameter d mounted on top (surmounted) or scooped out (depression):
Total Surface Area of Block with Surmounted Hemisphere:
Area = Area of all 6 faces of cube - Area of circular base of hemisphere
+ Curved Surface Area of hemisphere
= 6 * a^2 - pi * r^2 + 2 * pi * r^2
= 6 * a^2 + pi * r^2
Notice that whether the hemisphere is mounted on top or scooped out as a depression, the total surface area is IDENTICAL: the curved hemispherical bowl replaces the missing circular flat surface!
Chapter Exercises
Exercise 1: A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Exercise 2: A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Exercise 3: A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. (Take pi = 3.14).
Solutions and Explanations
Solution 1: The radius of both the cone and hemisphere is r = 3.5 cm = 7/2 cm. The total height of the toy is 15.5 cm. The height of the hemispherical base equals its radius: r = 3.5 cm. Therefore, the vertical height of the conical portion is h = Total Height - r = 15.5 - 3.5 = 12 cm. First find the slant height l of the cone: l = sqrt(r^2 + h^2) = sqrt((3.5)^2 + 12^2) = sqrt(12.25 + 144) = sqrt(156.25) = 12.5 cm. The total surface area of the toy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere: TSA = CSA(Cone) + CSA(Hemisphere) = pi * r * l + 2 * pi * r^2 = pi * r * (l + 2r) = (22/7) * (7/2) * (12.5 + 2(3.5)) = 11 * (12.5 + 7) = 11 * 19.5 = 214.5 cm^2. Thus, the total surface area of the toy is 214.5 cm^2.
Solution 2: The diameter of the capsule is d = 5 mm, so radius r = 5/2 = 2.5 mm. The total length of the capsule is 14 mm. The length of the cylindrical part is h = Total Length - 2r = 14 - 2(2.5) = 14 - 5 = 9 mm. The surface area of the capsule is: Surface Area = CSA(Cylinder) + 2 * CSA(Hemisphere) = 2 * pi * r * h + 2 * (2 * pi * r^2) = 2 * pi * r * (h + 2r). Notice that h + 2r = 9 + 5 = 14 mm (the total length). Substituting: Surface Area = 2 * (22/7) * (5/2) * 14 = (22/7) * 5 * 14 = 22 * 5 * 2 = 220 mm^2. Thus, the surface area of the capsule is 220 mm^2.
Solution 3: Diameter of base is 4 cm, so radius r = 2 cm. Height of cone is h = 2 cm. The volume of the toy is the sum of the volumes of the cone and the hemisphere: Volume(Toy) = (1/3)pir^2h + (2/3)pir^3 = (1/3)pir^2(h + 2r) = (1/3) * 3.14 * 2^2 * (2 + 2(2)) = (1/3) * 3.14 * 4 * 6 = 3.14 * 4 * 2 = 25.12 cm^3. Now consider a right circular cylinder circumscribing the toy: the cylinder must have the same radius r = 2 cm, and its height H must equal the total height of the toy: H = h + r = 2 + 2 = 4 cm. Volume of the cylinder is: Volume(Cylinder) = pi * r^2 * H = 3.14 * 2^2 * 4 = 3.14 * 16 = 50.24 cm^3. The difference in volumes is: Difference = Volume(Cylinder) - Volume(Toy) = 50.24 - 25.12 = 25.12 cm^3.