The volume of what? The material used to make the box? It's just the volume of the box minus the volume of the chambers. The volume of the box is 30 * 10 * 10, and the volume of each chamber is 4/3 * pi * r^3, where r = 5 because the diameter of each chamber is 10. So:
V = (30 * 10 * 10) - 3 * (4/3 * pi * 5^3)
V = 3000 - (3 * 4/3 * 125 * pi)
V = 3000 - 500 * pi
V = 3000 - 1570.80 (approx).
V = 1429.20 cm^3
The surface area is more ambiguous; since the spherical chambers are all within the box's interior, are they counted? If you split the box into a top half and a bottom half , then each half would have three hemispheric indentations as part of one of its sides, which is probably what's being asked. Let's consider that case for each half of the box, then multiply by two to get the total surface area of the two halves of the box.
Considering the bottom half of the box (as the top half is just a mirror image), the dimensions are 30 long * 10 wide * 5 high. So two of the sides are 30 * 5, two are 10 * 5, and the bottom is 30 * 10, so the total surface area of those five sides is: (2 * 30 * 5) + (2 * 10 * 5) + (30 * 10) = 300 + 100 + 300 = 700 cm^2.
The top is more complicated, because it's the surface area of the three hemispheres plus the surface area of the "flat" portion of the top. The area of the "flat" portion is 30 * 10, minus the area of the cross-sections of the three spheres at their widest point, which is just the area of three circles with diameter 10 and therefore raidus 5. So:
The area of the flat portion is (30 * 10) - 3 * pi * 5^2 = 300 - 75 * pi.
The surface area of the three hemispherical indentations is half of the surface area of three full spheres of the same radius, which would be 3 * (4 * pi * r^2). So, it's 3 * (2 * pi * 5^2) = 150 * pi.
So the total surface area of the top side is 300 - 75 * pi + 150 * pi, which is 300 + 75 * pi.
The total surface area of all six sides of the bottom half of the box is then: 700 + 300 + 75 * pi, which is 1000 + 75 * pi.
Therefore, the surface area of both halves of the box combined is just double that, or (2000 + 150 * pi) cm^2.
This is approximately 2471.24 cm^2.
So if you were building a box that's as described above, and which separates into equal top and bottom halves, you would need to mold it from 1429.20 cm^3 of material, and would need to paint 2471.24 cm^2 of surface area.