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Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 12: Q34. (page 480)

Suppose another box has a base with dimensions 8 by 6 and diagonal that forms a $70^{鈭�}$ angle with a diagonal of a base. Show that the ration of the volumes of the two boxes is $\frac{\tan 35^{鈭�}}{\tan 70^{鈭�}}$ .

Short Answer

Expert verified

The volume of two boxes is $\frac{\tan 35^{鈭�}}{\tan 70^{鈭�}}$ .

Step by step solution

01

Step 1. Given information.

A diagonal of a box forms a $35^{鈭�}$ angle with a diagonal of the base.

Another diagonal of a box forms a $70^{鈭�}$ angle with a diagonal of the base.

02

Step 2. Determine the height.

Use Pythagoras theorem,

$\begin{matrix} \sqrt{{(8)}^{2} + {(6)}^{2}} = d \\ d = \sqrt{64 + 36} [square] \\ d = 10 [simplify] \end{matrix}$

Now, the height of the boxes is,

$\begin{matrix} \tan 35^{鈭�} = \frac{h}{10} \\ h = 10 \tan 35^{鈭�} [crossmultiply] \end{matrix}$

And

$\begin{matrix} \tan 70^{鈭�} = \frac{h}{10} \\ h = 10 \tan 70^{鈭�} [crossmultiply] \end{matrix}$

03

Step 3. Determine the ratio of volume.

The volume of the box is:

$\begin{matrix} \frac{v_{1}}{v_{2}} = \frac{8 脳 6 脳 10 \tan 35^{鈭�}}{8 脳 6 脳 10 \tan 70^{鈭�}} \\ \frac{v_{1}}{v_{2}} = \frac{\tan 35^{鈭�}}{\tan 70^{鈭�}} \end{matrix}$

This is the required expression.

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