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Surface Areas and Volumes

Surface Areas and Volumes

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Hollow and Cut-out Solids
Accuracy: 0%
Total Attempted1
Total Correct0
Total Incorrect1
Difficulty Level:
High
Boundary Conditions in Composite Solids
Accuracy: 33%
Total Attempted3
Total Correct1
Total Incorrect2
Difficulty Level:
High
Mathematical Constraints in Volume Calculations
Accuracy: 47%
Total Attempted15
Total Correct7
Total Incorrect8
Difficulty Level:
High
Surface Area of Composite Solids
Accuracy: 65%
Total Attempted55
Total Correct36
Total Incorrect19
Difficulty Level:
Medium
Volume of Composite Solids
Accuracy: 71%
Total Attempted14
Total Correct10
Total Incorrect4
Difficulty Level:
Medium
Capacity and Displacement Problems
Accuracy: 75%
Total Attempted4
Total Correct3
Total Incorrect1
Difficulty Level:
Low
Curved Surface Area Calculations
Accuracy: 77%
Total Attempted26
Total Correct20
Total Incorrect6
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Calculate the total surface area of composite solids formed by combining basic shapes like cylinders, cones, and hemispheres, focusing on curved surface areas and boundary conditions.
  • Determine the volume of composite solids by summing the individual volumes of basic shapes, ensuring no overlap in calculations.
  • Derive and apply formulas for the curved surface areas of cylinders, cones, and hemispheres, using π\pi and specific dimensions.
  • Understand the impact of boundary conditions on surface area calculations, particularly where surfaces are internal or not visible in composite solids.
  • Apply mathematical constraints and assumptions in volume calculations to accurately represent the structure of combined solid shapes.

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Revision Notes & Summary

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Chapter Notes

Surface Area of Composite Solids

  • Total Surface Area (TSA): For a composite solid made by combining basic solids, the TSA is the sum of the curved surface areas (CSA) of the individual components that are visible.
  • Example: A cylinder with two hemispheres at its ends:
    • TSA = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere.

Volume of Composite Solids

  • Total Volume: The volume of a composite solid is the sum of the volumes of the individual solids.
  • Example: A cuboid surmounted by a half-cylinder:
    • Volume = Volume of cuboid + Volume of half-cylinder.

Curved Surface Area Calculations

  • Cylinder: CSA = 2πrh2\pi rh, where rr is the radius and hh is the height.
  • Cone: CSA = πrl\pi rl, where ll is the slant height.
  • Hemisphere: CSA = 2πr22\pi r^2, where rr is the radius.

Boundary Conditions in Composite Solids

  • When solids are joined, internal surfaces are not included in the TSA.
  • Example: A hemisphere attached to a cube does not include the base area of the hemisphere in the TSA.

Mathematical Constraints in Volume Calculations

  • Assumptions: Ensure no overlap in the volume of combined solids.
  • Example: A cylinder with a conical cavity:
    • Volume = Volume of cylinder - Volume of cone.

Examples

  • Example 1: A toy shaped like a cone on a hemisphere:
    • TSA = CSA of hemisphere + CSA of cone.
  • Example 2: A decorative block with a cube and hemisphere:
    • TSA = TSA of cube - base area of hemisphere + CSA of hemisphere.
  • Example 3: A rocket with a cone on a cylinder:
    • Orange area = CSA of cone + base area of cone - base area of cylinder.
    • Yellow area = CSA of cylinder + area of one base of the cylinder.
  • Example 4: A bird-bath shaped like a cylinder with a hemispherical depression:
    • TSA = CSA of cylinder + CSA of hemisphere.

Exercises

  • Exercise 1: Two cubes joined end to end, find the surface area of the resulting cuboid.
  • Exercise 2: A vessel with a hemispherical base and cylindrical top, find the inner surface area.
  • Exercise 3: A toy with a cone on a hemisphere, find the TSA.
  • Exercise 4: A cube with a hemisphere on top, find the maximum diameter of the hemisphere and TSA.
  • Exercise 5: A cube with a hemispherical depression, find the remaining surface area.
  • Exercise 6: A capsule with two hemispheres and a cylinder, find the surface area.

Summary

  • Surface Area: Combine the CSA of visible parts.
  • Volume: Sum the volumes of individual solids without overlap.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

12.56 cm³

B.

25.12 cm³

C.

37.68 cm³

D.

50.24 cm³
Correct Answer: B

Solution:

The volume of the toy is the sum of the volumes of the hemisphere and the cone. Volume of hemisphere = 23πr3\frac{2}{3} \pi r^3 and volume of cone = 13πr2h\frac{1}{3} \pi r^2 h. The radius r=2r = 2 cm. Volume of toy = 23×3.14×23+13×3.14×22×3=25.12\frac{2}{3} \times 3.14 \times 2^3 + \frac{1}{3} \times 3.14 \times 2^2 \times 3 = 25.12 cm³. The cylinder has a radius of 2 cm and height of 5 cm. Volume of cylinder = πr2h=3.14×22×5=62.8\pi r^2 h = 3.14 \times 2^2 \times 5 = 62.8 cm³. The difference in volume is 62.8−25.12=37.6862.8 - 25.12 = 37.68 cm³.

Chapter Concept:

Hollow and Cut-out Solids

A.

66.32 m²

B.

70.24 m²

C.

75.36 m²

D.

80.48 m²
Correct Answer: A

Solution:

The area of the canvas is the sum of the curved surface area of the cylinder and the curved surface area of the cone: 2πrh+πrl=2π×2×2.1+π×2×2.8=66.32 m22\pi rh + \pi rl = 2\pi \times 2 \times 2.1 + \pi \times 2 \times 2.8 = 66.32 \text{ m}^2.

Chapter Concept:

Surface Area of Composite Solids

A.

282.6 cm²

B.

314 cm²

C.

392.7 cm²

D.

424.8 cm²
Correct Answer: A

Solution:

The total surface area is the sum of the curved surface area of the cylinder and the two hemispheres: 2πrh+4πr2=2×3.14×2.5×9+4×3.14×2.52=282.62\pi rh + 4\pi r^2 = 2 \times 3.14 \times 2.5 \times 9 + 4 \times 3.14 \times 2.5^2 = 282.6 cm².

Chapter Concept:

Volume of Composite Solids

A.

63.585 cm²

B.

70 cm²

C.

80 cm²

D.

90 cm²
Correct Answer: A

Solution:

The area to be painted orange is the CSA of the cone plus the base area of the cone minus the base area of the cylinder.

Chapter Concept:

Volume of Composite Solids

A.

2πr2+πrl2\pi r^2 + \pi rl

B.

πr2+πrl\pi r^2 + \pi rl

C.

2πr2+2πrl2\pi r^2 + 2\pi rl

D.

πr2+2πrl\pi r^2 + 2\pi rl
Correct Answer: A

Solution:

The total surface area of the toy is the sum of the curved surface area of the hemisphere (2πr22\pi r^2) and the curved surface area of the cone (πrl\pi rl).

Chapter Concept:

Curved Surface Area Calculations

A.

50,000 g

B.

60,000 g

C.

70,000 g

D.

80,000 g
Correct Answer: B

Solution:

Calculate the volume of each cylinder and multiply by the density of iron to find the mass.

Chapter Concept:

Volume of Composite Solids

A.

CSA of hemisphere + CSA of cone

B.

CSA of hemisphere + TSA of cone

C.

TSA of hemisphere + CSA of cone

D.

TSA of hemisphere + TSA of cone
Correct Answer: A

Solution:

The total surface area of the toy is the sum of the curved surface area (CSA) of the hemisphere and the CSA of the cone.

Chapter Concept:

Volume of Composite Solids

A.

220 mm²

B.

250 mm²

C.

300 mm²

D.

350 mm²
Correct Answer: A

Solution:

The surface area of the capsule is the sum of the curved surface area of the cylinder and the curved surface areas of the two hemispheres.

Chapter Concept:

Surface Area of Composite Solids

A.

53π\frac{5}{3}\pi cm³

B.

43π\frac{4}{3}\pi cm³

C.

2π2\pi cm³

D.

π\pi cm³
Correct Answer: A

Solution:

The volume of the solid is the sum of the volumes of the cone and the hemisphere. Volume of cone = 13πr2h=13π(1)2(1)=13π\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (1)^2 (1) = \frac{1}{3}\pi cm³. Volume of hemisphere = 23πr3=23π(1)3=23π\frac{2}{3}\pi r^3 = \frac{2}{3}\pi (1)^3 = \frac{2}{3}\pi cm³. Total volume = 13π+23π=53π\frac{1}{3}\pi + \frac{2}{3}\pi = \frac{5}{3}\pi cm³.

Chapter Concept:

Mathematical Constraints in Volume Calculations

A.

150 kg

B.

160 kg

C.

170 kg

D.

180 kg
Correct Answer: B

Solution:

The volume of the first cylinder is πr2h=π(12)2×220\pi r^2 h = \pi (12)^2 \times 220. The volume of the second cylinder is π(8)2×60\pi (8)^2 \times 60. The total mass is 8×(π(12)2×220+π(8)2×60)=160 kg8 \times (\pi (12)^2 \times 220 + \pi (8)^2 \times 60) = 160 \text{ kg}.

Chapter Concept:

Surface Area of Composite Solids

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Surface Areas and Volumes

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True or False

Correct Answer: False

Solution:

The base area of the hemisphere is not included in the calculation of the total surface area, as it is not exposed when the hemisphere and cone are joined.

Chapter Concept :

Curved Surface Area Calculations

Correct Answer: False

Solution:

The internal surfaces where two solids join are not visible and hence not included in the total surface area calculation.

Chapter Concept :

Boundary Conditions in Composite Solids

Correct Answer: False

Solution:

The surface area of internal surfaces that are not visible is not included in the total surface area calculation of a composite solid.

Chapter Concept :

Boundary Conditions in Composite Solids

Correct Answer: False

Solution:

The base area of the hemisphere is not included in the total surface area calculation because it is covered by the cone, leaving only the curved surface areas exposed.

Chapter Concept :

Surface Area of Composite Solids

Correct Answer: False

Solution:

The surface area of a composite solid is not the simple sum of the surface areas of its components because some areas may overlap or be hidden.

Chapter Concept :

Volume of Composite Solids

Correct Answer: True

Solution:

When calculating the volume of a composite solid, the total volume is the sum of the volumes of the individual components, as there is no overlap in volume.

Chapter Concept :

Volume of Composite Solids

Correct Answer: True

Solution:

When calculating the volume of composite solids, the total volume is the sum of the individual volumes of the constituent solids, as there is no overlap in volume.

Chapter Concept :

Volume of Composite Solids

Correct Answer: True

Solution:

When calculating the volume of a composite solid, the volume is the sum of the volumes of its individual components, as no volume is lost in the process of combining them.

Chapter Concept :

Mathematical Constraints in Volume Calculations

Correct Answer: False

Solution:

When calculating the surface area of a composite solid, overlapping areas are not counted twice. Therefore, the total surface area is not simply the sum of the total surface areas of the individual components.

Chapter Concept :

Mathematical Constraints in Volume Calculations

Correct Answer: False

Solution:

The total surface area of a composite solid does not include the base area of the hemisphere when it is joined with another solid.

Chapter Concept :

Boundary Conditions in Composite Solids

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