A block of wood is in the shape of a rectangular prism with dimensions (n+5) cm, (n+8) cm, and (n+14) cm. All faces of the block are painted, and then the block is cut into 1 cm cubes using parallel cuts to each face. If exactly half of the cubes have no paint on them, find the total number of cubes.

The total number of cubes in the rectangular prism are 1,120 cubes

Step-by-step explanation:

The given parameters are;

The dimensions of the rectangular prism block of wood = (n + 5), (n + 8), and (n + 14)

The volume of the block of wood = (n + 5)×(n + 8)×(n + 14) = n³ + 27·n² + 222·n + 560

The quantity of the 1  cm cubes that have no paint = 1/2 of the total quantity = 1/2 the volume of the original cube

The cubes that have paint on them are the surface cubes, we have;

One layer of cube is removed from each of the eight sides, therefore, we have the dimensions of the cube rectangular prism that have no paint on them given as (n + 5 - 2), (n + 8 - 2), and (n + 14 - 2), which gives the dimensions as (n + 3), (n + 6), and (n + 12);

Therefore, we have

2 × (n + 3)×(n + 6)×(n + 12) = (n + 5)×(n + 8)×(n + 14)

2 × (n + 3)×(n + 6)×(n + 12) - (n + 5)×(n + 8)×(n + 14) = 0  

Which gives;

n³ + 15·n² + 30·n - 128 = 0

n = 2

n = -17/2 - √(33)/2 = -5.6277

n = √(33)/2 - 17/2 = -11.3722

Therefore, given that the smallest dimension of the painted prism = (n + 5), the possible value for n = 2

The volume of the rectangular prism = (n + 5) × (n + 8) × (n + 14) = (2 + 5) × (2 + 8) × (2 + 14) = 1120 cm³

Given that the cuts are 1 cm³ each, the total number of cubes in 1,120 cm³ = 1,120 cubes.