Question

A metal cuboid of dimension $22cm\times 15cm\times 7.5cm$ was melted and cast into a cylinder of height 14cm. What is its radius?

Answer 1

Hint: Focus on the point that the volume of the metal remains constant. The volume of the metal that we get by melting the cuboid is equal to the volume of the cuboid with the given dimensions. Just use the formulas that the volume of the cuboid is \[l\times b\times h\] and the volume of the cylinder is $\pi {{r}^{2}}h$ . Equate the two volumes and solve the equation to get the value of radius r.

Complete step by step solution:
Let us start by drawing a representative figure for better visualisation.

As it is given that the cuboid of the metal is melted and is cast into a cylinder, which means that the volume of metal in each case is equal. Also, we know that the volume of a cuboid is given by $l\times b\times h$ while the volume of a cylinder is equal to $\pi {{r}^{2}}h$ . It is also a fact that the volume of the cuboid is equal to the volume of metal, so the volume of the cylinder and cuboid must be equal. Also, all the dimensions of the cuboid are $22cm\times 15cm\times 7.5cm$ and the height of the cylinder is 14cm.
$l\times b\times h=\pi {{r}^{2}}h$
$\Rightarrow 22\times 15\times 7.5=\pi {{r}^{2}}\times 14$
Now we will put the value of $\pi $ as $\dfrac{22}{7}$ . On doing so, we get
$22\times 15\times 7.5=\dfrac{22}{7}\times {{r}^{2}}\times 14$
$\Rightarrow \dfrac{15\times 7.5}{2}={{r}^{2}}$
$\Rightarrow 7.5\times 7.5={{r}^{2}}$
Now if we take square root of both the sides of the equation, we get
$r=7.5cm$
Hence, the radius of the cylinder must be 7.5 cm.

Note: Make sure to convert all the dimensions to a standardized system of units; this decreases the chance of errors. It would also help if you remembered all the basic formulas for surface area and volume of the general 3-D shapes like the cone, cube, cylinder, etc. We can also multiply the dimensions of the cuboid and get the volume. Similarly, we could have calculated the volume of the cylinder in terms of r separately. Then, we could have equated them to get the value of r.