A large solid cube is melted and cast into 'N' small solid spheres, each of radius 3 cm, and 'N + 2' small solid cuboids, each of dimensions 4 cm $$\times$$ 4 cm $$\times$$ 6.5 cm. If the length of each side of the large solid cube is 12 cm, then find the value of 'N'. [Use $$\pi = \frac{22}{7}$$]

Solution

Volume of large solid cube = N $$\times$$ volume of small solid spheres + (N + 2) $$\times$$ volume of small solid cuboid

$$side^3\ =\ N\times\frac{4}{3}\times\pi\times radius^3\ +\left(N+2\right)\times\ length\times breadth\times\ height\ $$

$$12^3\ =\ N\times\frac{4}{3}\times\frac{22}{7}\times3^3\ +\left(N+2\right)\times\ 4\times\ 4\times\ 6.5$$

$$1728\ =\ N\times4\times\frac{22}{7}\times3^2\ +\left(N+2\right)\times104\ $$
$$1728\ =\ N\times\frac{88}{7}\times9\ +\left(N+2\right)\times104\ $$
$$864\ =\ N\times\frac{44}{7}\times9\ +\left(N+2\right)\times52\ $$
$$864\ =\ N\times\frac{396}{7}\ +52\ N+104$$
$$864-104=\frac{396N+364N}{7}$$
$$760=\frac{760N}{7}$$
N = 7