Question

66 cubic centimetres of silver is drawn into a wire 1mm in diameter. The length of the wire in meters will be
A .84m
B.90m
C.168m
D.336m

Answer 1

Hint:
Here we need to find the length of the wire in meters. We are provided with the measurement in cubic centimetre, which means the volume of the wire is given to us. We can obtain the length of the wire by substituting the values of volume and radius in the formula of volume of a cylinder. We will then convert the length from millimetre to metre.
Formula Used: We will use the formula of volume of cylinder,\[V = \pi {r^2}l\], where \[V\] is volume of the cylinder, \[r\] is radius of the cylinder and \[l\] is length of the cylinder.

Complete step by step solution:
We know that the volume of wire is equal to the volume of the cylinder. It is given that silver wire is drawn from 66 cubic centimetres of silver. So we can equate the volume of wire with 66 cubic centimetres.
So we can write \[{\text{volume of wire}} = \pi {r^2}l\].
The diameter is equal to 1 mm. Now we will find the radius of the wire using formula \[r = \dfrac{d}{2}\].
Substituting \[d = 1\] in the formula, we get
\[r = \dfrac{1}{2}{\text{mm}}\]
The volume is given in cubic centimetres. So to have the same unit we will convert cubic centimetre into cubic millimetres. We know that \[1{\text{c}}{{\text{m}}^3} = 1000{{\text{m}}^3}\], so the volume in cubic millimetre will be \[{\text{66c}}{{\text{m}}^3} = 66000{\text{m}}{{\text{m}}^3}\].
Substituting, \[V = 66000{\text{m}}{{\text{m}}^3}\] , \[{\text{r}} = \dfrac{1}{2}\] and \[\pi = \dfrac{{22}}{7}\] in the formula of volume of wire, we get \[\begin{array}{l}{\text{volume of wire}} = \pi {r^2}l\\66000 = \dfrac{{22}}{7} \times {\left[ {\dfrac{1}{2}} \right]^2} \times l\end{array}\]
Multiplying both the sides of equation by 7, we get
\[66000 \times 7 = 22 \times {\left[ {\dfrac{1}{2}} \right]^2} \times l\]
Dividing both side by 22, we get
\[\dfrac{{66000 \times 7}}{{22}} = \dfrac{1}{4} \times l\]
Now multiplying both side by 4, we get
\[\dfrac{{66000 \times 7 \times 4}}{{22}} = l\]
On simplifying the above equation, we get
\[l = 84000{\text{ mm}}\]
Now, we will convert the length in metre. As we know \[1000{\text{mm}} = 1{\text{m}}\], so we will divide the length by 1000 to convert its unit into meters.
\[\begin{array}{l}l = \dfrac{{84000}}{{1000}}{\text{m}}\\ = 84{\text{m}}\end{array}\]

\[\therefore\] We get the length as\[84{\text{m}}\]. Hence, the correct option is option [A].

Note:
Here, we need to be careful with the units, while solving the question. We need to remember that we can perform a mathematical operation on two or more numbers only if they have the same unit. If two numbers have different units then we have to convert either of them to the same unit as the other number. In this question, we were given diameter in millimetre and volume in cubic centimetre. So, we converted volume into cubic millimetre. We could have converted diameter in centimetre but it will be a decimal number so the calculation would have been a bit tough. It is very important for us to have knowledge of the conversion of units.