Question

The Volume of a pyramid with a square base is 200 cubic cm. The height of the pyramid is 13 cm. What will be the length of the slant edges (i.e. the distance between the apex and any other vertex), rounded to the nearest integer?

• A. 12 cm
• B. 13 cm
• C. 14 cm
• D. 15 cm
• E. 16 cm

Hint:

In square pyramids, the slant edge to a vertex is always calculated using the half diagonal of the base and the height. Always use Pythagoras theorem in 3D problems by breaking them into right triangles.

Answer 1

The Correct Option is C

Step 1: Recall volume formula of a square pyramid
The volume of a pyramid is given by: \[ V = \frac{1}{3} \times (\text{Base Area}) \times h \] Here, \(V = 200 \, \text{cm}^3\) and \(h = 13 \, \text{cm}\).

Step 2: Find the base area
\[ 200 = \frac{1}{3} \times (\text{Base Area}) \times 13 \] \[ \text{Base Area} = \frac{200 \times 3}{13} = \frac{600}{13} \approx 46.15 \, \text{cm}^2 \]

Step 3: Find the side of the square base
Let the side length be \(a\). Then: \[ a^2 = 46.15 \quad \Rightarrow \quad a = \sqrt{46.15} \approx 6.79 \, \text{cm} \]

Step 4: Find the slant edge (apex to vertex of base)
The apex is above the center of the base. The distance from the center of the square base to a vertex is half the diagonal: \[ \text{Half diagonal} = \frac{a \sqrt{2}}{2} = \frac{6.79 \times \sqrt{2}}{2} \approx \frac{6.79 \times 1.414}{2} \approx 4.80 \, \text{cm} \] Now, the slant edge is the hypotenuse of a right triangle with legs \(h = 13\) and half diagonal = 4.80. \[ \text{Slant edge} = \sqrt{13^2 + 4.80^2} = \sqrt{169 + 23.04} = \sqrt{192.04} \approx 13.86 \, \text{cm} \]

Step 5: Round to nearest integer
\[ \text{Slant edge} \approx 14 \, \text{cm} \] \[ \boxed{14 \, \text{cm}} \]