An icecream company makes a popular brand of icecream in a rectangular shaped bar 6 cm long 5 cm wide and 2 cm thick. To cut costs, the company has decided to reduce the volume of the box by `20%`. The thickness will remain the same but the length and width will be decreased by the same percentage amount. Which condition given below will the new length l satisfy ?

Options

A. `5.5lt l lt 6`
B. `5lt l lt 5.5`
C. `4.5 lt l lt 5`
D. `4 lt l lt 4.5`

Answer

To solve the problem step by step, we need to follow these calculations:

### Step 1: Calculate the original volume of the ice cream bar.
The volume \( V \) of a rectangular cuboid is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Length \( = 6 \, \text{cm} \)
- Width \( = 5 \, \text{cm} \)
- Height \( = 2 \, \text{cm} \)

Calculating the volume:
\[ V = 6 \times 5 \times 2 = 60 \, \text{cm}^3 \]

### Step 2: Calculate the new volume after reducing it by 20%.
To find the new volume after a 20% reduction:
\[ \text{New Volume} = \text{Original Volume} \times (1 - 0.20) = 60 \times 0.80 = 48 \, \text{cm}^3 \]

### Step 3: Set up the equation for the new dimensions.
Let the new length and width both decrease by \( x\% \). The new length \( L \) and new width \( W \) can be expressed as:
\[ L = 6 - \frac{6x}{100} = 6(1 - \frac{x}{100}) \]
\[ W = 5 - \frac{5x}{100} = 5(1 - \frac{x}{100}) \]
The height remains the same at \( 2 \, \text{cm} \).

### Step 4: Write the equation for the new volume.
The new volume can be expressed as:
\[ \text{New Volume} = L \times W \times \text{Height} = \left(6(1 - \frac{x}{100})\right) \times \left(5(1 - \frac{x}{100})\right) \times 2 \]
Setting this equal to the new volume:
\[ \left(6(1 - \frac{x}{100})\right) \times \left(5(1 - \frac{x}{100})\right) \times 2 = 48 \]

### Step 5: Simplify the equation.
Expanding the left-hand side:
\[ 60(1 - \frac{x}{100})^2 \times 2 = 48 \]
\[ 120(1 - \frac{x}{100})^2 = 48 \]
Dividing both sides by 120:
\[ (1 - \frac{x}{100})^2 = \frac{48}{120} = \frac{2}{5} \]

### Step 6: Solve for \( x \).
Taking the square root of both sides:
\[ 1 - \frac{x}{100} = \sqrt{\frac{2}{5}} \quad \text{or} \quad 1 - \frac{x}{100} = -\sqrt{\frac{2}{5}} \]
Since \( x \) must be a positive percentage, we only consider:
\[ 1 - \frac{x}{100} = \sqrt{\frac{2}{5}} \]
Thus:
\[ \frac{x}{100} = 1 - \sqrt{\frac{2}{5}} \]
\[ x = 100(1 - \sqrt{\frac{2}{5}}) \]

### Step 7: Find the new length \( L \).
Substituting \( x \) back into the expression for \( L \):
\[ L = 6(1 - \frac{x}{100}) = 6\sqrt{\frac{2}{5}} \]

### Conclusion
The new length \( L \) will satisfy the condition:
\[ L = 6\sqrt{\frac{2}{5}} \approx 6 \times 0.632 = 3.792 \, \text{cm} \]