/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Find the angle between a cube's ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 47

Find the angle between a cube's diagonal and one of its edges.

Short Answer

Expert verified
The angle between a cube's diagonal and one of its edges is approximately 54.74 degrees.

Step by step solution

01

Identify the right triangle

In a cube, a right triangle can be formed by one edge of the cube, the height from the edge to a vertex on the opposite face (which is also the length of a side since all edges of a cube are equal), and the diagonal connecting these two vertices.
02

Apply the concept of the trigonometric cosine function

The cosine of an angle in a right triangle is equal to the length of the adjacent side divided by the length of the hypotenuse. In this case, the length of the adjecent side is the edge of the cube, and the hypotenuse is the diagonal of the cube.
03

Calculate the length of the diagonal

By using Pythagoras theorem, you can find the length of the diagonal. Because a cube has all sides of equal length, if we assign \( a \) as the length of an edge, then the length of the diagonal \( d \) can be calculated as \( d = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3} \).
04

Apply the values to the cosine function

The cosine of the angle \( \theta \) between the diagonal and the edge of the cube is given by \( \cos(\theta) = \frac{a}{a\sqrt{3}} = \frac{1}{\sqrt{3}} \).
05

Find the angle

Finally, calculate the angle \( \theta \) by using the arccosine function to \( \frac{1}{\sqrt{3}} \). This gives an angle of approximately 54.74 degrees.

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Most popular questions from this chapter

Consider a regular tetrahedron with vertices \((0,0,0),(k, k, 0),(k, 0, k),\) and \((0, k, k),\) where \(k\) is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid \((k / 2, k / 2, k / 2)\) to two vertices. This is the bond angle for a molecule such as \(\mathrm{CH}_{4}\) or \(\mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron.

Use vectors to determine whether the points are collinear. (0,0,0),(1,3,-2),(2,-6,4)

Find the magnitude of \(v\). \(\mathbf{v}=\langle 1,0,3\rangle\)

Determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathbf{z}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j}+\frac{3}{4} \mathbf{k}\) (a) \(6 \mathbf{i}-4 \mathbf{j}+9 \mathbf{k}\) (b) \(-\mathbf{i}+\frac{4}{3} \mathbf{j}-\frac{3}{2} \mathbf{k}\) (c) \(12 \mathbf{i}+9 \mathbf{k}\) (d) \(\frac{3}{4} \mathbf{i}-\mathbf{j}+\frac{9}{8} \mathbf{k}\)

Find the magnitude of \(v\). Initial point of \(\mathbf{v}:(1,-3,4)\) Terminal point of \(\mathbf{v}:(1,0,-1)\)
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