To determine the amount of sheet metal needed to make a box with dimensions 12 inches by 18 inches by 30 inches, we need to calculate the surface area of the box.
Let's start by drawing a picture of the box:
```
+--------------------------+
/ /|
/ / |
/ / |
/ / |
/ / |
+--------------------------+ |
| | |
| | |
| | |
| | |
| | /
| | /
| | /
| | /
| | /
| |/
+--------------------------+
```
Now, let's break this down into individual faces. A rectangular box has six faces:
- Top and bottom face, each with dimensions 12 inches by 18 inches.
- Front and back face, each with dimensions 18 inches by 30 inches.
- Left and right face, each with dimensions 12 inches by 30 inches.
To find the surface area (total amount of sheet metal), we will calculate the area for each pair of faces and then sum them up.
1. Area of the top and bottom faces:
[tex]\[
\text{Area of one face} = 12 \times 18 = 216 \text{ square inches}
\][/tex]
Since there are two such faces, the total area for these two faces is:
[tex]\[
2 \times 216 = 432 \text{ square inches}
\][/tex]
2. Area of the front and back faces:
[tex]\[
\text{Area of one face} = 18 \times 30 = 540 \text{ square inches}
\][/tex]
Since there are two such faces, the total area for these two faces is:
[tex]\[
2 \times 540 = 1080 \text{ square inches}
\][/tex]
3. Area of the left and right faces:
[tex]\[
\text{Area of one face} = 12 \times 30 = 360 \text{ square inches}
\][/tex]
Since there are two such faces, the total area for these two faces is:
[tex]\[
2 \times 360 = 720 \text{ square inches}
\][/tex]
Finally, we sum up the areas of all the faces to get the total surface area of the box:
[tex]\[
432 + 1080 + 720 = 2232 \text{ square inches}
\][/tex]
Therefore, the amount of sheet metal needed to make the box is 2232 square inches.
