040 Review Problem - Volume comparisons of two water crystals

Problem 40
When a liquid freezes, it changes into a mass of solid bodies, each of which has a definite geometric form and is known as a crystal. Crystals are always bounded by plane surfaces, which are arranged in an orderly fashion with reference to imaginary lines drawn through the crystal and called its area. Every crystal has therefore a definite geometric form. Compare the volumes of crystals, one in the shape of a hexagonal right prism 'a' units on a side and '2a' units in altitude, and the other in the shape of two oppositely directed hexagonal regular pyramids, each 'a' units high with a common base 'a' units on a side.

Solution 40

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$A_b = 6 \times \frac{1}{2}a^2 \sin 60^\circ$

$A_b = \frac{3\sqrt{3}}{2} a^2 ~ \text{unit}^2$

Crystal prism (h₁ = 2a units)
$V_1 = A_b h_1 = \frac{3\sqrt{3}}{2} a^2(2a)$

$V_1 = 3\sqrt{3} \, a^3 ~ \text{unit}^3$

Crystal pyramid (h₂ = a units)
$V_2 = 2 \times \frac{1}{3}A_b h_2 = 2 \times \frac{1}{3}(\frac{3\sqrt{3}}{2} a^2)(a)$

$V_2 = \sqrt{3} \, a^3 ~ \text{unit}^3$

$\dfrac{V_1}{V_2} = \dfrac{3\sqrt{3} \, a^3}{\sqrt{3} \, a^3}$

$V_1 = 3V_2$

Thus, the volume of crystal prism is three times the volume of the crystal pyramid. answer

040 Review Problem - Volume comparisons of two water crystals

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