Answer :
To find the approximate change in the volume of a cube when its sides are increased by 1%, we can follow a series of logical steps.
### Step 1: Initial Volume Calculation
First, let's assume the initial side length of the cube is [tex]\( x \)[/tex] meters.
The volume of a cube is given by:
[tex]\[ V_{\text{initial}} = x^3 \][/tex]
### Step 2: New Side Length Calculation
If each side of the cube is increased by 1%, the new side length becomes:
[tex]\[ x_{\text{new}} = x \times 1.01 \][/tex]
### Step 3: New Volume Calculation
With the new side length, the volume of the cube can be calculated as:
[tex]\[ V_{\text{new}} = (x \times 1.01)^3 \][/tex]
### Step 4: Change in Volume Calculation
The change in the volume ([tex]\(\Delta V\)[/tex]) is the difference between the new volume and the initial volume:
[tex]\[ \Delta V = V_{\text{new}} - V_{\text{initial}} \][/tex]
### Step 5: Percentage Increment in Volume
The percentage increment in the volume is calculated by:
[tex]\[ \text{Percentage Increment} = \left(\frac{\Delta V}{V_{\text{initial}}}\right) \times 100 \][/tex]
### Applying Values
- Initial Volume: Assume [tex]\( x = 1 \)[/tex] meter.
[tex]\[ V_{\text{initial}} = 1^3 = 1 \text{ cubic meter} \][/tex]
- New Side Length:
[tex]\[ x_{\text{new}} = 1 \times 1.01 = 1.01 \text{ meters} \][/tex]
- New Volume:
[tex]\[ V_{\text{new}} = (1.01)^3 = 1.030301 \text{ cubic meters} \][/tex]
- Change in Volume:
[tex]\[ \Delta V = 1.030301 - 1 = 0.030301 \text{ cubic meters} \][/tex]
- Percentage Increment:
[tex]\[ \text{Percentage Increment} = \left(\frac{0.030301}{1}\right) \times 100 = 3.0301\% \][/tex]
### Conclusion
The approximate change in the volume of a cube with a side [tex]\( x \)[/tex] meters, when the sides are increased by 1%, is approximately [tex]\( 0.030301 \)[/tex] cubic meters. The percentage increment in the volume is approximately [tex]\( 3.0301\% \)[/tex].
### Step 1: Initial Volume Calculation
First, let's assume the initial side length of the cube is [tex]\( x \)[/tex] meters.
The volume of a cube is given by:
[tex]\[ V_{\text{initial}} = x^3 \][/tex]
### Step 2: New Side Length Calculation
If each side of the cube is increased by 1%, the new side length becomes:
[tex]\[ x_{\text{new}} = x \times 1.01 \][/tex]
### Step 3: New Volume Calculation
With the new side length, the volume of the cube can be calculated as:
[tex]\[ V_{\text{new}} = (x \times 1.01)^3 \][/tex]
### Step 4: Change in Volume Calculation
The change in the volume ([tex]\(\Delta V\)[/tex]) is the difference between the new volume and the initial volume:
[tex]\[ \Delta V = V_{\text{new}} - V_{\text{initial}} \][/tex]
### Step 5: Percentage Increment in Volume
The percentage increment in the volume is calculated by:
[tex]\[ \text{Percentage Increment} = \left(\frac{\Delta V}{V_{\text{initial}}}\right) \times 100 \][/tex]
### Applying Values
- Initial Volume: Assume [tex]\( x = 1 \)[/tex] meter.
[tex]\[ V_{\text{initial}} = 1^3 = 1 \text{ cubic meter} \][/tex]
- New Side Length:
[tex]\[ x_{\text{new}} = 1 \times 1.01 = 1.01 \text{ meters} \][/tex]
- New Volume:
[tex]\[ V_{\text{new}} = (1.01)^3 = 1.030301 \text{ cubic meters} \][/tex]
- Change in Volume:
[tex]\[ \Delta V = 1.030301 - 1 = 0.030301 \text{ cubic meters} \][/tex]
- Percentage Increment:
[tex]\[ \text{Percentage Increment} = \left(\frac{0.030301}{1}\right) \times 100 = 3.0301\% \][/tex]
### Conclusion
The approximate change in the volume of a cube with a side [tex]\( x \)[/tex] meters, when the sides are increased by 1%, is approximately [tex]\( 0.030301 \)[/tex] cubic meters. The percentage increment in the volume is approximately [tex]\( 3.0301\% \)[/tex].
