A 4cm x 5cm x 7cm rectangular prism is painted on all faces. If the prism is cut into 1cm x 1cm x 1cm cubes, how many cubes do not have paint on any of its sides?
To find out how many cubes do not have paint on any of its sides, we first need to calculate the total number of cubes in the rectangular prism.
To do this, we need to find the volume of the prism, which can be calculated by multiplying the length, width, and height.
Volume = (length) x (width) x (height)
Volume = 4cm x 5cm x 7cm
Volume = 140 cubic cm
Since each of the 1cm x 1cm x 1cm cubes was created by dividing the prism, we can now calculate how many of these cubes can be created.
140 cubic cm / (1cm x 1cm x 1cm) = 140 cubes
Therefore, there are 140 cubes in total.
Now, let's find out how many cubes have paint on at least one of their sides.
Since all faces of the prism are painted, we can exclude the cubes that are part of the surface.
The surface area of the prism can be calculated by finding the sum of the area of each face.
Surface Area = 2(lw + lh + wh)
Surface Area = 2(4cm x 5cm + 4cm x 7cm + 5cm x 7cm)
Surface Area = 2(20cm2 + 28cm2 + 35cm2)
Surface Area = 2(83cm2)
Surface Area = 166cm2
Since each face of the rectangular prism has an area of 166cm2, and each cube has an area of 1cm2, we can calculate the number of cubes that have paint on at least one of their sides.
Cubes with paint = Surface Area / (1cm x 1cm)
Cubes with paint = 166cm2 / (1cm x 1cm)
Cubes with paint = 166 cubes
Therefore, there are 166 cubes with paint on at least one of their sides.
Finally, to find the number of cubes that do not have paint on any of their sides, we subtract the number of painted cubes from the total number of cubes.
Cubes without paint = Total cubes - Cubes with paint
Cubes without paint = 140 cubes - 166 cubes
Cubes without paint = -26 cubes
Since we cannot have negative numbers of cubes, it means that all the cubes have paint on at least one of their sides.
Therefore, there are no cubes that do not have paint on any of its sides.
all the outer cubelets have paint. So, reduce the dimensions by 2 to exclude the outer faces, and the inner (paintless) solid is
2x3x5
So, 30 cubelets have no paint.