A rectangular prism has a volume of 162 cubic centimeters. If the length, width, and height are reduced to 1/3 of their original size what will the new volume be
the volume of similar objects is proportional to the cube of their corresponding sides, so
V1 : V2 = 1^3 : 3^3
V1 / 162 = 1/27
V1 = 162/27 = 6
the new volume is 6 cm^3
To find the new volume of the rectangular prism after reducing the length, width, and height by 1/3, you can follow these steps:
1. Start by finding the original volume of the rectangular prism, which is given as 162 cubic centimeters.
2. Let's assume the original length, width, and height of the rectangular prism are represented as L, W, and H, respectively.
3. Use the formula for volume of a rectangular prism: Volume = Length × Width × Height.
4. Substitute the given volume of 162 cubic centimeters into the formula: 162 = L × W × H.
5. Next, reduce the length, width, and height by 1/3. This means each of them will become 1 - (1/3) = 2/3 of their original size.
6. Now, note that the new length, width, and height are (2/3) × L, (2/3) × W, and (2/3) × H, respectively.
7. Plug these new values into the volume formula: New Volume = (2/3)L × (2/3)W × (2/3)H.
8. Simplify the equation:
New Volume = (2/3) × (2/3) × (2/3) × L × W × H.
New Volume = (8/27) × L × W × H.
9. Recall that the original volume was 162 cubic centimeters, so substitute this value into the equation:
162 = (8/27) × L × W × H.
10. To find the new volume, solve the equation for L × W × H:
(8/27) × L × W × H = 162.
11. Multiply both sides of the equation by (27/8) to isolate L × W × H:
L × W × H = (27/8) × 162.
12. Calculate the right-hand side of the equation:
L × W × H = 729.
Therefore, the new volume of the rectangular prism after reducing the length, width, and height by 1/3 will be 729 cubic centimeters.