Question

A stand is a frustum shape, formed by removing a small square-based pyramid from the top of
a large square-based pyramid.
The wood has a density of 85 g/cm3.
Calculate the mass of the stand in g.
9 cm height
4 cm
10 cm base

Answer 1

The frustum can be considered as consisting of a square pyramid, less the

cut volume.

• The mass of the stand, is approximately 18.60905 kg

Reasons:

The given parameter of the frustum are;

The density of the frustum, ρ = 85 g/cm³

Height of pyramid, h = 9 cm

Side length of base = 10 cm

Height of frustum

By proportional shapes, the side length of the top of the frustum can be found as follows;

$\dfrac{4 \, cm}{9 \, cm} = \dfrac{x}{10 \, cm}$

$x = \dfrac{4 \, cm}{9 \, cm} \times 10 \, cm = 4\frac{4}{9} \, cm$

$V = \dfrac{h}{3} \times \left(B_1 + B_2 + \sqrt{B_1 \times B_2} \right)$

B₁ = 10², B₂ = $\left(4\frac{4}{9} \right)^2$

Therefore;

$V = \dfrac{4}{3} \times \left(10^2 + \left(4\frac{4}{9} \right)^2 + \sqrt{10^2 \times \left(4\frac{4}{9} \right)^2} \right) \approx 218.93$

The volume of the stand, V ≈ 218.93 cm³

Mass = Volume × Density

∴ Mass of the stand, m = 218.93 cm³ × 85 g/cm³ = 18,609.05 grams = 18.60905 kg.

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