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Difficulty: MediumVolume and Surface Area of 3D Figures

An architectural firm is analyzing the material requirements and interior spaces of several proposed structures. Match each structural description to its correct calculated capacity (volume) or exterior area (surface area).

  • The interior capacity (volume) of a rectangular room measuring 1010 feet by 1212 feet with an 88-foot ceiling.960960 cubic feet
  • The total exterior area (surface area) of a cubic storage pod with an edge length of 1010 feet.600600 square feet
  • The volume of a cylindrical water tank with a base radius of 55 feet and a height of 2020 feet.500π500\pi cubic feet
  • The surface area of a spherical decorative sculpture with a radius of 1010 feet.400π400\pi square feet

Answer

The rectangular room matches 960960 cubic feet, the cubic pod matches 600600 square feet, the cylindrical tank matches 500π500\pi cubic feet, and the spherical sculpture matches 400π400\pi square feet.
Each structural description correctly aligns with its corresponding formula and numerical result as detailed in the solution steps.

Step-by-Step Solution

1
Calculate the volume of the rectangular room.
V=l×w×h=10×12×8=960V = l \times w \times h = 10 \times 12 \times 8 = 960 cubic feet.
The capacity of a rectangular room is found using the volume formula for a rectangular prism.
2
Calculate the surface area of the cubic storage pod.
SA=6×s2=6×(10)2=6×100=600SA = 6 \times s^2 = 6 \times (10)^2 = 6 \times 100 = 600 square feet.
A cube has 6 identical square faces, each with an area of s2s^2.
3
Calculate the volume of the cylindrical water tank.
V=π×r2×h=π×(5)2×20=25×20×π=500πV = \pi \times r^2 \times h = \pi \times (5)^2 \times 20 = 25 \times 20 \times \pi = 500\pi cubic feet.
The volume of a cylinder is the area of its circular base times its height.
4
Calculate the surface area of the spherical sculpture.
SA=4×π×r2=4×π×(10)2=4×100×π=400πSA = 4 \times \pi \times r^2 = 4 \times \pi \times (10)^2 = 4 \times 100 \times \pi = 400\pi square feet.
The surface area formula for a sphere is 44 times the area of a circle with the same radius.

Key Concept

Applying appropriate volume and surface area formulas for fundamental 3D figures: rectangular prisms, cubes, cylinders, and spheres.
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Topics

Volume and Surface Area of 3D Figures

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