Question

What length of solid cylinder \[2cm\] in diameter must be taken to recast into a hollow cylinder of external diameter \[20cm\], \[0.25cm\] thick and \[15cm\] long?
a). \[54.06cm\]
b). \[74.06cm\]
c). \[34.06cm\]
d). \[64.06cm\]

Answer 1

Hint: Here we are asked to find the length or height of a solid cylinder of diameter \[2cm\] to recast a hollow cylinder of \[20cm\] diameter, \[0.25cm\] thick and \[15cm\] long. Since the solid cylinder is going to recast into a hollow cylinder their volume will be equal. Thus, we will find the volume of both solid and hollow cylinders using the formula and then by equating both we will find the height or length of a solid cylinder (since that will be an unknown).
Formula Used: Formulas that we need to know before solving this problem:
The volume of a solid cylinder \[ = \pi {r^r}h\] where \[r\] the radius of the cylinder, \[h\] the height of the cylinder.
The volume of a hollow cylinder \[ = \pi \left( {{r_{ex}}^2 - {r_{in}}^2} \right)h\] where \[{r_{ex}}\] the external radius of a cylinder, \[{r_{in}}\] the internal radius of a cylinder, and \[h\] the height of a cylinder.

Complete step-by-step solution:
It is given that a hollow cylinder of external diameter \[20cm\], \[0.25cm\] thick and \[15cm\] long has been re-casted from a solid cylinder of diameter \[2cm\] and length or height \[h\]. We aim to find the height of the solid cylinder.
Let us first find the volume of a solid cylinder. It is given that the diameter of a solid cylinder is \[2cm\].
Thus, the radius of a solid cylinder is \[1cm\] and we don’t know the height of the solid cylinder so we will keep it as \[h\].
We know that the volume of a solid cylinder \[ = \pi {r^r}h\] is \[r\] the radius of the cylinder, \[h\] the height of the cylinder.
Therefore, the volume of the given solid cylinder \[{V_s} = \pi \times {\left( 1 \right)^2} \times h\]
On simplifying this we get \[{V_s} = \pi h..............(1)\]
We got the volume of a solid cylinder now let us find the volume of a hollow cylinder.
The given hollow cylinder has an external diameter \[20cm\], \[0.25cm\] thick, and \[15cm\] height.
We know that volume of a hollow cylinder \[ = \pi \left( {{r_{ex}}^2 - {r_{in}}^2} \right)h\] where \[{r_{ex}}\] the external radius of a cylinder, \[{r_{in}}\] the internal radius of a cylinder, and \[h\] the height of a cylinder thus, the volume of this hollow cylinder is \[{V_h} = \pi \times \left( {{r_{ex}}^2 - {r_{in}}^2} \right) \times 15\]
But we don’t know the values of the external and internal radius of the hollow cylinder. Let's find them.
We know that the thickness of a hollow cylinder is \[0.25cm\] and the external diameter \[20cm\] thus, the external radius \[{r_{ex}}\] is \[10cm\].
Therefore, the internal radius \[{r_{in}} = \left( {10 - 0.25} \right)cm = 9.75cm\]
Now let us substitute the radii in the hollow cylinder volume formula.
\[2cm\]
On simplifying this we get
\[{V_h} = \pi \times \left( {100 - 95.0625} \right) \times 15\]
\[{V_h} = \pi \times \left( {4.9375} \right) \times 15\]
\[{V_h} = 74.0625\pi ............(2)\]
Thus, we have found the volume of a hollow cylinder. It is given that the solid cylinder is re-casted into a hollow cylinder therefore, the volume of both cylinders will be equal.
Hence, \[(1) = (2)\]
That is \[{V_s} = {V_h}\]
\[ \Rightarrow \pi h = 74.0625\pi \], where \[h\] is the height of a solid cylinder
On simplifying this we get
\[ \Rightarrow h = 74.0625\]
Rounding off the above value to two decimal places we get
\[h = 74.06cm\]
Thus, a solid cylinder of length \[74.06cm\]and diameter \[2cm\] is needed to make a hollow cylinder of external diameter \[20cm\], \[0.25cm\] thick and \[15cm\] long. Now let us see the options to find the correct one.
Option (a) \[54.06cm\] is an incorrect option as we got the value of the height of a solid cylinder as\[h = 74.06cm\]
Option (b) \[74.06cm\] is the correct answer as we got the same value in our calculation above.
Option (c) \[34.06cm\] is an incorrect option as we got the value of the height of a solid cylinder as\[h = 74.06cm\]
Option (d) \[64.06cm\]is an incorrect option as we got the value of the height of a solid cylinder as\[h = 74.06cm\]
Option (b) \[74.06cm\] is the correct option.

Note: Recast is the process of melting a material (any metal) to produce a new shape. Rounding off a decimal number can be done in the following way: In a decimal number, consider the rightmost digit (The last digit after the decimal point) and check whether it is greater than or equal to five if yes, drop that digit and add one to the preceding digit otherwise just drop the digit and don’t add one to the preceding digit. This step is repeated till we reach the decimal place we require.