Question

How many shots each having diameter $ 3cm $ , can be made from a cuboidal lead solid of dimensions $ 9cm * 11cm * 12cm $ ?

Answer 1

Hint: Here we are given a cuboidal lead solid and the lead solid is changed into spherical lead shots, and then we need to find the number of shots that can be made. To find the required answer, we need the formula to find the volume of the cuboid and the volume of the sphere. Also, we need to equate both volumes to obtain the desired answer.

Formula to be used:
a) \[Volume{\text{ }}of{\text{ }}cuboid = length \times breadth \times height\] (Here length, breadth, and height are the dimensions of the cuboid)
b) \[Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{4}{3}\pi {r^3}\] Here $ r $ is the radius of the sphere.

Complete step by step answer:
First, we need to note the given information.
The length of the cuboid is $ 12cm $ , the breadth of the cuboid is $ 11cm $ and the height of the cuboid is $ 9cm $ .
Also, the diameter of the sphere is $ 3cm $ .
We all know that the radius is half the diameter. Thus the required radius of the sphere is $ r = \dfrac{3}{2} = 1.5cm $ .
Now, we shall find the volume of the cuboid.
\[Volume{\text{ }}of{\text{ }}cuboid = length \times breadth \times height\].
Thus, \[Volume{\text{ }}of{\text{ }}cuboid = 12 \times 11 \times 9\]
Hence we get\[Volume{\text{ }}of{\text{ }}cuboid = 1188c{m^3}\].
Now, we shall find the volume of the sphere.
\[Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{4}{3}\pi {r^3}\]
Thus \[Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{4}{3} \times \dfrac{{22}}{7} \times {\left( {1.5} \right)^3}\]
 $ \Rightarrow Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{4}{3} \times \dfrac{{22}}{7} \times \dfrac{3}{2} \times \dfrac{3}{2} \times \dfrac{3}{2} $
 $ \Rightarrow Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{{11}}{7} \times 3 \times 3 $
 $ \Rightarrow Volume{\text{ }}of{\text{ }}the{\text{ }}sphere = \dfrac{{99}}{7}c{m^3} $
We are asked to find the number of shots that can be made when the lead solid is changed into lead shots.
Let $ n $ be the number of lead shots.
Hence we need to multiply the number of shots with the volume of the sphere and need to compare it with the volume of the cuboid.
That is Volume of $ n $ spherical lead shots $ = $ Volume of the cuboid
 $ \Rightarrow n\dfrac{{99}}{7} = 1188 $
 $ \Rightarrow n = 1188 \times \dfrac{7}{{99}} $
 $ \Rightarrow n = 7 \times 12 $
 $ \Rightarrow n = 84 $
Therefore $ 84 $ shots can be made.

Note:

    The dimensions of the cuboid are $ 9cm * 11cm * 12cm $ . We may get confusions to choose the length, breadth, and height from the given $ 9cm * 11cm * 12cm $ . When we analyze the figure of the cuboid, we are able to note that the length is the largest dimension and height is the smallest dimension. So, we need to assume the largest value as the length and the smallest value as the breadth but the answer of this question will not be different if we choose dimensions either way.