Q33. - Geometry

Question

A diagonal of a box forms a ${35}^{\circ }$ angle with a diagonal of the base. Use trigonometry to approximate the volume of the box.

Step-by-Step Solution

Answer

The volume of box is$336.1{\text{\hspace{0.17em}\hspace{0.17em}unit}}^3$ .

Step 1. Given information.

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

The figure provided in the question,

Step 2. Determine the height.

Use Pythagoras theorem,

$\begin{array}{c}\sqrt{{\left(8\right)}^2+{\left(6\right)}^2}=d\\ d=\sqrt{64+36}\text{ [square]}\\ d=10\text{ [simplify]}\end{array}$

Now, the height of the prism is,

$\begin{array}{c}\tan {35}^{\circ }=\frac{h}{10}\\ h=10\tan {35}^{\circ }\text{ [cross multiply]}\end{array}$

Step 3. Determine the total volume.

The volume of the box is:

$\begin{array}{c}v=8\times 6\times 10\tan {35}^{\circ }\\ =336.1\text{ [simplify]}\end{array}$