Given:
The volume of a prism with equilateral triangular cross-section is 270cm³.
Length of the prism = [tex]10\sqrt{3}[/tex] cm
To find:
The length of the side of the equilateral triangular cross-section.
Solution:
Formulae used:
Area of an equilateral triangle is
[tex]Area=\dfrac{\sqrt{3}}{4}a^2[/tex]
Where a is the side length of equilateral triangle.
Volume of prism is
[tex]V=Bh[/tex]
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 [tex]\dfrac{\sqrt{3}}{4}a^2[/tex] sq. cm.
The volume of the prism is
[tex]V=\dfrac{\sqrt{3}}{4}a^2\times 10\sqrt{3}[/tex]
[tex]270=\dfrac{10(3)}{4}a^2[/tex]
[tex]270=\dfrac{30}{4}a^2[/tex]
[tex]270=7.5a^2[/tex]
Divide both sides by 7.5.
[tex]\dfrac{270}{7.5}=a^2[/tex]
[tex]36=a^2[/tex]
[tex]\pm \sqrt{36}=a[/tex]
[tex]6=a[/tex]
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.
