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 = $10\sqrt{3}$ cm

To find:

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

Solution:

Formulae used:

Area of an equilateral triangle is

$Area=\dfrac{\sqrt{3}}{4}a^2$

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  $\dfrac{\sqrt{3}}{4}a^2$ sq. cm.

The volume of the prism is

$V=\dfrac{\sqrt{3}}{4}a^2\times 10\sqrt{3}$

$270=\dfrac{10(3)}{4}a^2$

$270=\dfrac{30}{4}a^2$

$270=7.5a^2$

Divide both sides by 7.5.

$\dfrac{270}{7.5}=a^2$

$36=a^2$

$\pm \sqrt{36}=a$

$6=a$

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.