Question

the volume of a prism with equilateral triangular cross-section is 270cm^3 . If the length if the prism is 10 root 3 cm, what is the length of the side of the equilateral triangular cross-section ?​

Answer 1

Given:

The volume of a prism with equilateral triangular cross-section is 270cm³.

Length of the prism =
`10√(3)` cm

To find:

The length of the side of the equilateral triangular cross-section.

Solution:

Formulae used:

Area of an equilateral triangle is

`Area=(√(3))/(4)a^2`

Where a is the side length of equilateral triangle.

Volume of prism is

`V=Bh`

Where, B is base area and h is the height of the triangular prism.

Cross section of the prism is an equilateral triangular so the base area of the prism is
`(√(3))/(4)a^2` sq. cm.

The volume of the prism is

`V=(√(3))/(4)a^2* 10√(3)`

`270=(10(3))/(4)a^2`

`270=(30)/(4)a^2`

`270=7.5a^2`

Divide both sides by 7.5.

`(270)/(7.5)=a^2`

`36=a^2`

`\pm √(36)=a`

`6=a`

It takes only positive value because the side cannot be negative.

Therefore, the length of the side of the equilateral triangular cross-section is 6 cm.