If length, breadth and height of a cuboid is increased by x%, y% and z% respectively then its volume is increased by

Options

A. `[x+y+z+(xy+xz+yz)/(100)+(xyz)/((100)^(2))]%`
B. `[x+y+z+(xy+xz+yz)/(100)]%`
C. `[x+y+z+(xyz)/((100)^(2))]%`
D. None of these

Answer

To find the percentage increase in the volume of a cuboid when its length, breadth, and height are increased by x%, y%, and z% respectively, we can follow these steps:

### Step 1: Define the Original Dimensions
Let the original dimensions of the cuboid be:
- Length (L)
- Breadth (B)
- Height (H)

### Step 2: Calculate the New Dimensions
When the dimensions are increased by x%, y%, and z%, the new dimensions become:
- New Length = \( L + \frac{x}{100}L = L(1 + \frac{x}{100}) \)
- New Breadth = \( B + \frac{y}{100}B = B(1 + \frac{y}{100}) \)
- New Height = \( H + \frac{z}{100}H = H(1 + \frac{z}{100}) \)

### Step 3: Calculate the Original Volume
The original volume (V) of the cuboid is given by:
\[ V = L \times B \times H \]

### Step 4: Calculate the New Volume
The new volume (V') after the increases is:
\[ V' = \text{New Length} \times \text{New Breadth} \times \text{New Height} \]
\[ V' = L(1 + \frac{x}{100}) \times B(1 + \frac{y}{100}) \times H(1 + \frac{z}{100}) \]
\[ V' = L \times B \times H \times (1 + \frac{x}{100})(1 + \frac{y}{100})(1 + \frac{z}{100}) \]

### Step 5: Expand the New Volume
Using the formula for the product of three binomials, we can expand \( (1 + \frac{x}{100})(1 + \frac{y}{100})(1 + \frac{z}{100}) \):
\[ V' = V \times \left(1 + \frac{x+y+z}{100} + \frac{xy + xz + yz}{10000} + \frac{xyz}{1000000}\right) \]

### Step 6: Calculate the Change in Volume
The change in volume (ΔV) is:
\[ \Delta V = V' - V \]
\[ \Delta V = V \left( \left(1 + \frac{x+y+z}{100} + \frac{xy + xz + yz}{10000} + \frac{xyz}{1000000}\right) - 1 \right) \]
\[ \Delta V = V \left( \frac{x+y+z}{100} + \frac{xy + xz + yz}{10000} + \frac{xyz}{1000000} \right) \]

### Step 7: Calculate the Percentage Increase in Volume
To find the percentage increase in volume, we use:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V} \right) \times 100 \]
\[ \text{Percentage Increase} = \left( \frac{\frac{x+y+z}{100} + \frac{xy + xz + yz}{10000} + \frac{xyz}{1000000}}{1} \right) \times 100 \]
\[ \text{Percentage Increase} = x + y + z + \frac{xy + xz + yz}{100} + \frac{xyz}{10000} \]

### Final Result
Thus, the volume of the cuboid is increased by:
\[ x + y + z + \frac{xy + xz + yz}{100} + \frac{xyz}{10000} \]