Question

If the volume of an equilateral triangle based prism is \[72\sqrt{3\,}{{m}^{3}}\] and height is 7m, find the length of the base.

Answer 1

Hint: We will first use the volume of prism formula \[V=Bh\] to find the area of the base from it and then we will equate it with the area of the equilateral triangle and then we will be able to find the length of the base of the prism.

Complete step-by-step answer:
The general formula for the volume of the prism is \[V=Bh.....(1)\] where B is the area of the base and h is the height.
So it is mentioned in the question that the volume of the prism is \[72\sqrt{3\,}{{m}^{3}}\] and the height is 7m. So substituting these two values in equation (1) we get the area of the base as,
\[72\sqrt{3}=B\times 7.....(2)\]
So now solving for B in equation (2) we get,
\[B=\dfrac{72\sqrt{3}}{7}\,{{m}^{2}}.....(3)\]
The base of the prism is a triangle and we know the area of the triangle and hence it will be the area of the base as well. So using this information we get,
Area of the base(B) \[=\dfrac{\sqrt{3}}{4}{{s}^{2}}...........(4)\] where s is the length of the base.
So now substituting the value of B from equation (3) in equation (4) we get,
\[\Rightarrow \dfrac{72\sqrt{3}}{7}=\dfrac{\sqrt{3}}{4}{{s}^{2}}...........(5)\]
Now cancelling the similar constants in equation (5) we get,
\[\Rightarrow \dfrac{72}{7}=\dfrac{1}{4}{{s}^{2}}...........(6)\]
Now cross multiplying and isolating s in equation (6) we get,
\[\Rightarrow {{s}^{2}}=\dfrac{72\times 4}{7}...........(7)\]
Now taking square root on both sides of the equation (7) we get,
\[\Rightarrow s=\sqrt{\dfrac{72\times 4}{7}}=\sqrt{\dfrac{288}{7}}=\dfrac{12\sqrt{2}}{\sqrt{7}}m\]
Hence the length of the base of the prism is \[\dfrac{12\sqrt{2}}{\sqrt{7}}m\].

Note: Remembering the formula of the volume of the equilateral triangle based prism is the key here and also remembering the formula of the area of the equilateral triangle is important. We in a hurry can make a mistake in equation (5) by writing 2 in place of 4 in the denominator and hence we need to be careful while doing this step.