Chapter 12: Q34. (page 480)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

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

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,

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

And

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

Step 3. Determine the ratio of volume.

The volume of the box is:

$\begin{matrix} \frac{v_{1}}{v_{2}} = \frac{8 \times 6 \times 10 \tan 35^{\circ}}{8 \times 6 \times 10 \tan 70^{\circ}} \\ \frac{v_{1}}{v_{2}} = \frac{\tan 35^{\circ}}{\tan 70^{\circ}} \end{matrix}$

This is the required expression.

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the height.

Step 3. Determine the ratio of volume.

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Suppose another box has a base with dimensions 8 by 6 and diagonal that forms a70∘angle with a diagonal of a base. Show that the ration of the volumes of the two boxes is tan35∘tan70∘.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the height.

Step 3. Determine the ratio of volume.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Geometry

Study anywhere. Anytime. Across all devices.

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