Question

If the volume of a right equilateral prism is \[8500\sqrt 3 {\rm{ d}}{{\rm{m}}^3}\] whose height is 50 cm, then find the side of the base.
(a) \[10\sqrt {17} \] cm
(b) \[10\sqrt {17} \] dm
(c) \[20\sqrt {17} \] cm
(d) \[20\sqrt {17} \] dm

Answer 1

Hint:
Here, we need to find the length of the side of the base of the prism. We will assume \[a\] to be the length of the side of the base of the right equilateral prism. We will use the formula for the area of an equilateral triangle in the formula for the volume of the prism, and obtain an equation in terms of \[a\]. Then, we will solve the equation to get the value of \[a\], and hence, find the length of the side of the base.

Formula Used:
We will use the formula of the area of an equilateral triangle is given by the formula \[\dfrac{{\sqrt 3 }}{4}{a^2}\], where \[a\] is the length of the side of the triangle.

Complete step by step solution:
First, we will convert all the units to decimetre.
We know that 1 decimetre \[ = \] 10 centimetre.
Multiplying both sides of the equation by 5, we get
5 decimetre \[ = \] 50 centimetre
Therefore, the height of 50 cm is equal to 5 dm.
Now, let \[a\] be the length of the side of the base of the right equilateral prism.
We will use the formula for volume of a prism to form an equation in terms of \[a\].
The volume of a prism is the product of the area of its base and the height of the prism.
The area of the base of the right equilateral prism is the area of the equilateral triangle.
We know that the area of an equilateral triangle is given by the formula \[\dfrac{{\sqrt 3 }}{4}{a^2}\].
The height of the right equilateral prism is 5 dm.
Therefore, we get
Volume of right equilateral prism \[ = \dfrac{{\sqrt 3 }}{4}{a^2} \times 5\]
It is given that the volume of the right equilateral prism is \[8500\sqrt 3 {\rm{ d}}{{\rm{m}}^3}\].
Thus, we get the equation
\[ \Rightarrow 8500\sqrt 3 = \dfrac{{\sqrt 3 }}{4}{a^2} \times 5\]
We need to solve this equation to find the value of \[a\] and hence, find the length of the side of the base.
Simplifying the equation, we get
\[ \Rightarrow 8500\sqrt 3 = \dfrac{{5\sqrt 3 }}{4}{a^2}\]
Dividing both sides of the equation by \[\sqrt 3 \], we get
\[ \Rightarrow 8500 = \dfrac{5}{4}{a^2}\]
Multiplying both sides of the equation by \[\dfrac{4}{5}\], we get
\[ \Rightarrow 8500 \times \dfrac{4}{5} = \dfrac{5}{4}{a^2} \times \dfrac{4}{5}\]
Simplifying the expression, we get
\[ \Rightarrow 6800 = {a^2}\]
Taking the square root on both the sides, we get
\[ \Rightarrow \sqrt {{a^2}} = \sqrt {6800} \]
Simplifying the expression, we get
\[ \Rightarrow a = \sqrt {400 \times 17} {\rm{ dm}}\]
Thus, we get
\[ \Rightarrow a = 20\sqrt {17} {\rm{ dm}}\]
Therefore, we get the length of the side of the base of the right equilateral prism as \[20\sqrt {17} \] dm.

Thus, the correct option is option (d).

Note:
We have formed a linear equation in one variable using the given information in this question. A linear equation in one variable is an equation which has only one variable with highest exponent 1 and is of the form \[ax + b = 0\], where \[a\] and \[b\] are integers. A linear equation of the form \[ax + b = 0\] has only one solution. Also, we need to be careful while applying the formula, as wrong formula will give us the wrong result. As the base of an equilateral prism is similar to the shape of an equilateral triangle so we used the formula of area of equilateral triangle.