Q. 58 - Calculus

Question

Some crystals have rhombohedral structures. A rhombohedron is a parallelepiped in which all of the edge lengths are equal and each of the six faces is a congruent rhombus. Find the volumes of the rhombohedral crystals described in Exercises 58 and 59.

Each side length is 1 cm, and the acute angles in each face measure ${60}^o$. (Hint: Let vector i form one of the edges of the rhombohedron, and let a second nonparallel edge be in the $xy$-plane.)

Step-by-Step Solution

Answer

The volumes of the rhombohedral crystals is $0.705$ cube unit.

Step 1. Given Information

Some crystals have rhombohedral structures. A rhombohedron is a parallelepiped in which all of the edge lengths are equal and each of the six faces is a congruent rhombus. We have to find the volumes of the rhombohedral crystals.

Each side length is 1 cm, and the acute angles in each face measure ${60}^{◦}$. 

Step 2. We have to find the volumes of the rhombohedral crystals.

$V=a^3(1-\cos \theta )\sqrt{1+2\cos \theta }$

From the question $a=1 \mathrm{and} \theta ={60}^o$

$$\begin{gathered} V={\left(1\right)}^3(1-\cos {60}^o)\sqrt{1+2\cos {60}^o} \\ V=1\left(1-\frac{1}{2}\right)\sqrt{1+2\times \frac{1}{2}} \\ V=\left(\frac{2-1}{2}\right)\sqrt{1+1} \\ V=\left(\frac{1}{2}\right)\sqrt{2} \\ V=0.5\times 1.41 \\ V=0.705 \end{gathered}$$