Certainly! Let's break down the problem step-by-step.
1. Given:
- Side length of the cube-shaped box: 15 inches.
- Number of identical cube-shaped blocks inside the box: 27.
2. Finding the side length of one block:
- Since the box is a cube and there are 27 smaller cubes inside it, each side of the box must be divisible by the side length of the smaller cubes.
- The number 27 can be expressed as [tex]\(3^3\)[/tex], so there will be 3 smaller cubes along each dimension of the box.
- Therefore, the side length of one smaller cube = side length of the box / 3.
- Calculating the side length: [tex]\(15 / 3 = 5\)[/tex] inches.
So, the side length of one block is 5 inches.
3. Finding the total surface area of the 27 blocks:
- Surface area of one smaller cube = 6 × (side length of block)^2.
- Plugging in the side length: [tex]\(6 × (5)^2 = 6 × 25 = 150\)[/tex] square inches for one block.
- Since there are 27 blocks, total surface area = [tex]\(27 × 150 = 4050\)[/tex] square inches.
So, the total surface area of the 27 blocks is 4050 square inches.
4. Finding the surface area of the box:
- Surface area of the box = 6 × (side length of box)^2.
- Plugging in the side length: [tex]\(6 × (15)^2 = 6 × 225 = 1350\)[/tex] square inches.
So, the surface area of the box is 1350 square inches.
Now that we have all the necessary values, we can fill in the blanks in the originally given question:
"The side length of the blocks is 5 inches, so the total surface area of the 27 blocks is 4050 square inches, compared to the surface area of the box, which is 1350 square inches."
Thus, compared to the surface area of the box, the total surface area of the 27 blocks is significantly larger.
The completed text would be:
"The side length of the blocks is 5 inches, so the total surface area of the 27 blocks is 4050 square inches, compared to the surface area of the box, which is 1350 square inches."
