Question

What is the volume of a solid metal cylinder of height 4 centimetres and radius 5 centimetres? This solid is melted and recast into 5 cones of equal height and radius 2 centimetres. Find the height of such a cone.

Answer 1

Hint: We should know the formula to calculate the volume of cone if radius and height of the cylinder r were given. The volume of the cylinder is equal to \[\pi {{r}^{2}}h\]. Now it was given that the solid is melted into 5 cones of equal height and radius of 2 cm. We should remember that the volume of substance will remain constant after recasting. By using this concept we can find the height of a cone.

Complete step by step answer:
Before solving the question, we should know that the volume of a 3D figure is equal to the product of area of base of cone and height of base of cone.

So, by using this concept we can get the volume of the cylinder by this concept.
We know that the cylinder will have a circle of radius r as the base of the cylinder.
So, the area of the base of the cylinder of radius r is equal to \[\pi {{r}^{2}}\].
Let us assume the height of the cylinder is equal to h.
So, the volume of the cylinder is equal to \[\pi {{r}^{2}}h\].
From the question,
We were given that the height of the cylinder is equal to 4 cm and radius of base of cylinder is equal to 5 cm.
Let the volume of the cylinder whose base radius equal to 5 cm and height is equal to 4 cm is equal to V.
So,
\[\begin{align}
  & \Rightarrow V=\pi {{(5)}^{2}}(4) \\
 & \Rightarrow V=100\pi \\
\end{align}\]
Hence, the volume of the cylinder is equal to \[100\pi \].
We know that a cone occupies \[{{\dfrac{1}{3}}^{rd}}\]of volume of a cylinder of radius r and height h. So, the volume of a cone is equal to \[\dfrac{1}{3}\pi {{r}^{2}}h\].

From the question, it was given that the cylinder is divided into 5 cones of equal height and radius 2 cm.
Let us assume the height of a cone is equal to h.
So, the volume of one cone \[=\dfrac{1}{3}\pi {{(2)}^{2}}h=\dfrac{4\pi h}{3}\]
So, the volume of 5 identical cones \[=5\left( \dfrac{4\pi h}{3} \right)=\dfrac{20\pi h}{3}\]
We are given that a cylinder is casted into 5 cones. So, the volume of the cylinder is equal to 5 times the volume of the cone.
So, we get
\[\Rightarrow \dfrac{20\pi h}{3}=100\pi \]
By using cross multiplication, we get
\[\begin{align}
  & \Rightarrow 20\pi h=300\pi \\
 & \Rightarrow h=15cm \\
\end{align}\]
So, the height of each cone is equal to 15 cm.

Note: Some students will have a misconception that after recasting the surface area of the solid may remain constant. If this misconception is followed, we will get wrong results. This type of misconception should be avoided to avoid getting wrong results. There may be a chance of having calculation mistakes. So, a student should be careful while solving this problem.