Math Problem Statement

ABCDE is a square-based pyramid. The vertex E is directly above the center of the face ABCD. AB = 16 cm and AE = 20 cm. Calculate the following: a) The angle between line AE and the plane face ABCD. b) The angle between plane BCE and face ABCD. c) The angle between lines AE and EC. d) The volume of the pyramid. e) The total surface area of the pyramid.

Solution

Let's go through the problem step by step based on the given pyramid structure. Here's how we can address each part:

a) Angle between line [AE] and the plane face ABCD:

  • Since the vertex EE is directly above the center DD of the square base ABCDABCD, the line AEAE is perpendicular to the base plane ABCDABCD. Therefore, the angle between AEAE and the plane ABCDABCD is 90°.

b) Angle between the plane face BCE and the face ABCD:

  • To find the angle between the two planes, we need to use the vector normal to each plane.
  • Plane BCEBCE: The vectors to consider are from BB to CC and from BB to EE.
  • Plane ABCDABCD: Since it's a horizontal plane, the normal vector is vertical (i.e., along line DEDE).
  • The angle between the planes can be calculated using the dot product of these normal vectors.

c) Angle between the lines [AE] and [EC]:

  • Using the triangle AECAEC, we can apply the cosine rule or simple trigonometry to calculate the angle.
  • The length of ACAC can be calculated using the Pythagorean theorem, since ABCDABCD is a square with side length 16 cm.

d) Find the volume of the pyramid:

  • The formula for the volume of a pyramid is: V=13×Base Area×HeightV = \frac{1}{3} \times \text{Base Area} \times \text{Height}
    • Base area: ABCDABCD is a square with side length 16 cm, so the area is 162=256cm216^2 = 256 \, \text{cm}^2.
    • Height: This is the vertical distance from EE to the base, which can be determined using Pythagoras' theorem in triangle AEDAED. h=AE2AD2h = \sqrt{AE^2 - AD^2}

e) Find the total surface area of the pyramid:

  • The surface area is the sum of the area of the base and the areas of the four triangular faces.
    • Base area = 256cm2256 \, \text{cm}^2.
    • For each triangular face, the area can be calculated using the formula: A=12×Base×Slant HeightA = \frac{1}{2} \times \text{Base} \times \text{Slant Height}
      • Use AB=16cmAB = 16 \, \text{cm} and the slant height for the side triangles.

Would you like to go into further detail for any part of this solution? Here's a set of follow-up questions:

  1. How do we apply the dot product to find the angle between two planes?
  2. Can you show the steps for calculating the slant height in triangle BCE?
  3. How do we apply the cosine rule in triangle AEC?
  4. How is the height of the pyramid derived from the triangle AED?
  5. What is the exact method for calculating the total surface area, including the lateral faces?

Tip: Always visualize the geometry in 3D when solving pyramid-related problems to understand angles and dimensions better.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Geometry
Vectors

Formulas

Volume of a pyramid: V = (1/3) × Base Area × Height
Surface area of a pyramid = Base area + Sum of the triangular face areas
Pythagorean theorem: a^2 + b^2 = c^2
Cosine rule: cos(θ) = (b² + c² - a²) / (2bc)

Theorems

Pythagoras' Theorem
Cosine Rule
Dot Product for angle between planes

Suitable Grade Level

Grades 10-12