A square of side 5 cm is cut off from each corner of rectangular sheet of paper having length to width ratio of 11:7. It is now used to make a box with a base having length to width ratio of 9:5. The volume of the box so formed is (in `cm^(3)`)

Options

A. 19250
B. 9625
C. 8750
D. 5625

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the dimensions of the rectangular sheet
Given the length to width ratio of the rectangular sheet is 11:7, we can denote:
- Length = 11x
- Width = 7x

### Step 2: Calculate the dimensions after cutting corners
Since we are cutting squares of side 5 cm from each corner, the dimensions of the remaining shape will be:
- New Length = 11x - 10 (5 cm from each end)
- New Width = 7x - 10 (5 cm from each end)

### Step 3: Set up the ratio for the base of the box
The base of the box has a length to width ratio of 9:5, which can be expressed as:
- Length of the base = 9y
- Width of the base = 5y

### Step 4: Relate the dimensions of the base to the remaining dimensions
From the remaining dimensions we have:
- Length: 11x - 10 = 9y
- Width: 7x - 10 = 5y

### Step 5: Solve the equations
We can set up the equations:
1. \( 11x - 10 = 9y \)
2. \( 7x - 10 = 5y \)

From the second equation, we can express y in terms of x:
\[ y = \frac{7x - 10}{5} \]

Substituting this into the first equation:
\[ 11x - 10 = 9 \left(\frac{7x - 10}{5}\right) \]

### Step 6: Clear the fraction and simplify
Multiply both sides by 5 to eliminate the fraction:
\[ 5(11x - 10) = 9(7x - 10) \]
\[ 55x - 50 = 63x - 90 \]

### Step 7: Solve for x
Rearranging gives:
\[ 55x - 63x = -90 + 50 \]
\[ -8x = -40 \]
\[ x = 5 \]

### Step 8: Calculate the dimensions of the base
Now substituting \( x = 5 \) back into the equations for the dimensions:
- Length = \( 11(5) - 10 = 55 - 10 = 45 \) cm
- Width = \( 7(5) - 10 = 35 - 10 = 25 \) cm

### Step 9: Determine the height of the box
The height of the box is equal to the side of the square cut from the corners, which is 5 cm.

### Step 10: Calculate the volume of the box
The volume \( V \) of the box can be calculated using the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Height} \]
\[ V = 45 \times 25 \times 5 \]

Calculating this gives:
\[ V = 1125 \times 5 = 5625 \, \text{cm}^3 \]

### Final Answer
The volume of the box formed is **5625 cm³**.

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