Answer to Question #145387 in Geometry for fred

Question

A monument is a stone structure in the form of a frustum of a regular square pyramid whose height is 50m and whose base
edges are 4.5m and 10m respectively. Through the center of the monument is a circular opening 3m in diameter at the top and
6m in diameter at the bottom.
1. What is the slant height of the opening?
2. What is the lateral surface area of the opening?
3. Find the volume of stone in the monument.

Answer 1

1. Using concept of similar triangles:

(Height of top part of pyramid/length of smaller base edge) = (Total height of pyramid/length of bigger base edge)

(x / 4.5) = ((50 + x) / 10)

Therefore, x = 40.9m

This implies that total height of pyramid = 50 + 40.9 = 90.9m

To get slant height,

Slant height of opening = Slant height of pyramid - Slant height of truncated top

To get length of adjacent sides of base edges:

4.5m base:

4.5² = a² + a²

4.5² = 2a²

a = (9√2)/4

10m base

10² = b² + b²

100 = 2b²

b = 5√2

Slant height of pyramid,l₁

(l₁ ) ²= 90.9² + (5√2)²

l₁ = 91.17m

Slant height of truncated top, l₂

(l₂)² = 40.9² + ((9√2)/4)²

l₂ = 41.02m

Therefore, slant height of opening = 91.17 - 41.02 = 50.15m

2) Lateral surface area of opening = Area of big circular opening + Area of small circular + 4(Area of triangle 1 - Area of triangle 2)

=πR² + πr² + 4(T_{1 +} T₂)

=π(R² + r²) + 4(T₁ + T₂)

=π(3² + 1.5²) + 4((0.5 * B * H') - (0.5 * b * h'))

where B and b = base of triangles

H' and h' = slant heights of triangles

= (45π/4) + 4((0.5 * 10 * 91.17) - (0.5 * 4.5 * 41.02))

=1489.56m³

3. 3) Volume of stone in the monument = ((1/3) * B * H) - ((1/3) * b * h)

Where B and b = base areas

H and h = heights

B = 10 * 10 = 100m²

b = 4.5 * 4.5 = 20.25m²

=((1/3) * 100 * 90.9) - ((1/3) * 20.25 * 41.02)

=2753.115m³