To find the total surface area of the rectangular solid, you first need to determine the dimensions of the solid using the given volume and the ratio of the length, width, and height.
1. Let's assume the length, width, and height of the rectangular solid are 3x, 2x, and x respectively (as they are in the ratio of 3:2:1).
2. The volume of a rectangular solid is given by the formula: Volume = length × width × height. Given that the volume is 384 cm³, you can set up the equation as follows:
3x * 2x * x = 384
6x² * x = 384
6x³ = 384
x³ = 64
x = 4
3. Now that you have found the value of x, you can determine the dimensions of the rectangular solid:
Length = 3x = 3 * 4 = 12 cm
Width = 2x = 2 * 4 = 8 cm
Height = x = 4 cm
4. To find the total surface area of the box, you need to calculate the sum of the areas of all six faces of the rectangular solid. The formula for the total surface area of a rectangular solid is:
Total Surface Area = 2lw + 2lh + 2wh
Substituting the values of length, width, and height:
Total Surface Area = 2(12)(8) + 2(12)(4) + 2(8)(4)
Total Surface Area = 192 + 96 + 64
Total Surface Area = 352 cm²
Therefore, the total surface area of the rectangular solid is 352 cm².
