Question

The radius of a sphere is 9 cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire in meters.
1) 972
2) 792
3) 292
4) 97.2

Answer 1

Hint: Since, the sphere is melted and converted into wire, the volume of the sphere and wire will be equal. Therefore, we first find the volume of the sphere using the formula $\dfrac{4}{3}\pi {r^3}$, where $r$ is the radius of the sphere. Next, use this as the volume of the cylinder to find the length of the wire.
Substitute the values of volume and radius of wire in the formula, where \[{r_1}\] is the radius of the wire, to find the length of the wire.

Complete step by step answer:

Since all of the material of the sphere is used to form the wire (considering no losses ) the volume of the sphere must be equal to the volume of the wire.
We shall first calculate the volume of the sphere of radius 9 cm. The volume of a sphere with radius $r$ is given by $\dfrac{4}{3}\pi {r^3}$.
On substituting the value of $r$in the formula of volume of the sphere, we get,
$
  V = \dfrac{4}{3}\pi {r^3} \\
  V = \dfrac{4}{3}\pi {\left( 9 \right)^3} \\
  V = 972\pi {\text{c}}{{\text{m}}^3} \\
 $
The wire with the circular cross-section can be treated as a cylinder. The volume of a cylinder with cross-sectional radius of ${r_1}$ and length $l$ is given by \[\pi r_1^2l\]
Let the length of the wire be $l$ and diameter be 2 mm that is equivalent to 0.2cm.
As the radius is half the diameter, the radius of the wire will be 0.1cm
The volume of the wire will be :
$
  V = \pi {\left( {0.1} \right)^2}l \\
  V = \left( {0.01} \right)\pi l{\text{ c}}{{\text{m}}^3} \\
$
So, volume of the wire is $V = \left( {0.01} \right)\pi l{\text{ c}}{{\text{m}}^3}$
Since the volume of the sphere and the volume of the wire are equal, we can solve for the length of the wire.
$
  972\pi = \left( {0.01} \right)\pi l \\
  \dfrac{{972}}{{0.01}} = l \\
  l = 97200{\text{ cm}} \\
$
But, we have to calculate the answer in m. So, divide the length by 100 to convert the unit in m.
\[\dfrac{{97200}}{{100}}{\text{ m}} = 972{\text{ m}}\]
Hence, option A is the correct answer.


Note: The shape of the wire is a cylinder and not just a line. Whenever a thing is melted into another, the volume remains the same. The conversion of the unit plays an important role in this question. Also, we have to convert the length into m after dividing by 100.