Question

The length, breadth and height of a rectangular cuboid are in the ratio 1: 2 : 3. If the length, breadth and height are increased by 100\(\%\) each then what would be the increase in the volume of the cuboid ?

• A. 7 times
• B. 8 times
• C. 9 times
• D. 10 times

Answer 1

The Correct Option is A

To determine the increase in the volume of the cuboid when the dimensions are increased by 100%, we need to perform the following calculations step-by-step:

1. Given the initial ratio of dimensions: length : breadth : height = 1 : 2 : 3. 
2. Let's assume the original dimensions of the cuboid are:
   • Length \( l = x \)
   • Breadth \( b = 2x \)
   • Height \( h = 3x \)
3. Original Volume \( V_{original} \):
   \(V_{original} = l \times b \times h = x \times 2x \times 3x = 6x^3\)
4. When increased by 100%, the new dimensions become:
   • New Length = \( 2x \)
   • New Breadth = \( 4x \)
   • New Height = \( 6x \)
5. New Volume \( V_{new} \):
   \(V_{new} = (2x) \times (4x) \times (6x) = 48x^3\)
6. The increase in volume is calculated as follows:
   \(V_{increase} = V_{new} - V_{original} = 48x^3 - 6x^3 = 42x^3\)
7. To find out how many times the increase is relative to the original volume, calculate:
   \(\frac{V_{new}}{V_{original}} = \frac{48x^3}{6x^3} = 8\)
8. Thus, the volume of the cuboid is 8 times the original volume. Therefore, the increase is \(8 - 1 = 7\) times the original volume.
9. Correct answer: 7 times.