The base of a pyramid is a rectangle whose length and breadth are 16 cm and 12 cm respectively. All lateral edges from the vertex to the base vertices are 26 cm long, and the pyramid is a right rectangular pyramid. What is the volume of the pyramid in cubic centimetres?

Introduction / Context:
This question involves a three dimensional solid called a right rectangular pyramid. The base is a rectangle, and all lateral edges from the vertex to the base vertices are equal, which simplifies some geometry. The task is to find the volume of this pyramid. Problems like this combine knowledge of the Pythagoras theorem in space with the volume formula for pyramids and are common in more advanced aptitude and engineering exams.

Given Data / Assumptions:
• Base is a rectangle with length 16 cm and breadth 12 cm.• All lateral edges (from vertex to each base vertex) are 26 cm long.• The pyramid is a right rectangular pyramid, so the vertex lies vertically above the centre of the base.• We need the volume of the pyramid.

Concept / Approach:
The volume V of a pyramid is V = (1/3) * base area * height. The base area is easy to compute as length times breadth. The challenge is to find the vertical height of the pyramid. Because it is a right rectangular pyramid, the foot of the perpendicular from the vertex to the base is at the centre of the rectangle. The line from the centre of the base to any vertex of the base, together with the vertical height and the lateral edge, forms a right triangle. Using Pythagoras theorem in this triangle allows us to find the height.

Step-by-Step Solution:
Step 1: Compute the base area.Base area = length * breadth = 16 * 12 = 192 cm^2.Step 2: Find the distance from the centre of the rectangle to a base vertex.The rectangle diagonal = sqrt(16^2 + 12^2) = sqrt(256 + 144) = sqrt(400) = 20 cm.The centre is at the midpoint of the diagonal, so the distance from the centre to a vertex is half of 20, which is 10 cm.Step 3: Use Pythagoras theorem in the triangle formed by the height, this 10 cm segment, and the lateral edge.Let h be the vertical height and the lateral edge be 26 cm.Then 26^2 = h^2 + 10^2.676 = h^2 + 100.h^2 = 676 − 100 = 576.h = sqrt(576) = 24 cm.Step 4: Compute the volume of the pyramid.V = (1/3) * base area * height = (1/3) * 192 * 24.192 * 24 = 4608.V = 4608 / 3 = 1536 cubic centimetres.

Verification / Alternative check:
We can double check the arithmetic: 192 * 24 can be computed as 200 * 24 − 8 * 24 = 4800 − 192 = 4608. Dividing 4608 by 3 gives 1536, since 3 * 1500 = 4500 and 3 * 36 = 108, which sums to 4608. The height 24 cm is also reasonable because it is slightly less than the lateral edge 26 cm but significantly larger than the base radius 10 cm, consistent with a right pyramid geometry.

Why Other Options Are Wrong:
Option B: 1024 is too small and would arise only if either the height or base area were miscalculated.Option C: 718 does not match the required one third factor and suggests multiple arithmetic errors.Option D: 2072 is too large compared to the correct volume and contradicts the relation V = (1/3) * base area * height.

Common Pitfalls:
Students often mistake the lateral edge for the vertical height and directly use 26 cm as height, which gives an incorrect volume. Others miscalculate the distance from the centre of the base to a vertex or fail to divide by 3 in the volume formula for a pyramid. Drawing a clear diagram of the pyramid and marking the right triangles involved helps in avoiding these errors.

Final Answer:
The volume of the pyramid is 1536 cubic centimetres.