Q. 59 - 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 2 cm, and the acute angles in each face measure ${45}^o$. (See the hint in the previous exercise.)

Step-by-Step Solution

Answer

The volumes of the rhombohedral crystals is $3.6$ cubic centimetre .

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 ${45}^{◦}$.

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 $a=2 \mathrm{and} \theta ={45}^o$

$$\begin{gathered} V={\left(2\right)}^3(1-\cos {45}^o)\sqrt{1+2\cos {45}^o} \\ V=8\left(1-\frac{1}{\sqrt{2}}\right)\sqrt{1+2\times \frac{1}{\sqrt{2}}} \\ V=8\left(1-\frac{1}{1.41}\right)\sqrt{1+\sqrt{2}} \\ V=8\left(1-0.71\right)\sqrt{1+1.41} \\ V=8\times 0.29\sqrt{2.41} \\ V=8\times 0.29\times 1.55 \\ V=3.596 \\ V=3.6 \end{gathered}$$