Question

How many spherical balls can be made out of a solid cube of lead whose edge measure 44 cm and each ball being 4 cm in diameter.

Answer 1

A cube is a three-dimensional object having all the sides of equal length with 8 vertices and 12 edges. All the sides are parallel and perpendicular to each other. A sphere is also a three-dimensional object with a center and the radius as the measuring parameter.
In this question, we need to determine the number of the spherical balls that can be made from the cube of edge length 44 cm. For this to be done, we have to equate the volume of the cube with the product of the volume of one spherical ball and the total number of spherical balls.

Complete step by step answer:
The volume of the cube is given as ${V_c} = {a^3}$ where a is the measurement of the side of the cube.
Substitute $a = 44{\text{ cm}}$ in the formula ${V_c} = {a^3}$ to determine the volume of the solid cube from which small spherical balls are to be made.

${V_c} = {a^3} \\\ = {(44)^3} \\\ = 85184{\text{ c}}{{\text{m}}^3} - - - - (i) \\\$

Now, the volume of the sphere is given as ${V_s} = \dfrac{4}{3}\pi {r^3}$ where r is the radius of the sphere.
Substitute $r = 4{\text{ cm}}$ in the formula ${V_s} = \dfrac{4}{3}\pi {r^3}$ to determine the volume of one spherical ball.
${V_s} = \dfrac{4}{3}\pi {r^3} \\\ = \dfrac{4}{3}\pi {(4)^3} \\\ = \dfrac{4}{3} \times \dfrac{{22}}{7} \times 64 \\\ = 268.19{\text{ c}}{{\text{m}}^3} - - - - (ii) \\\$

According to the question, the cube of the side’s length 44 cm has been melted to build spherical balls of radius 4 cm. Let us consider that the total number of small spherical balls made be $x$. So, ${V_c} = x{V_s} - - - - (iii)$

Now, substitute the values of the volume of the cube and the volume of the spherical ball from equation (i) and (ii) respectively in the equation (iii) to determine the total number of spherical balls.

${V_c} = x{V_s} \\\ 84184 = x \times 268.19 \\\ x = \dfrac{{84184}}{{268.19}} \\\ = 313.89 \\\$

As the number of the total spherical balls is needed so, fractions should be neglected. Hence, the total number of spherical balls of radius 4 cm made up from the cube of side 44 cm is 313.

Note: Students should note here that the number of the objects (here, spherical balls) should always be in the whole number and not in fractions. However, if we round off the number 313.89 then, we will get 314, but the raw material of the melted cube is fixed, and so it cannot be exceeded.