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Question

A metallic sheet of rectangular shape is 48 cm × 36 cm. From each corner a square of side x cm is cut off and an open box is made of the remaining sheet. If the box holds 5.12 litre of water, then what is the value of x ?

This question was previously asked in
CDS 1 2026 Maths Question Paper (12-Apr-2026)
The correct answer is
8 cm

Metallic Sheet Box Volume Problem

This problem involves calculating the dimension 'x' of squares cut from a metallic sheet to form an open box of a given volume.

Box Formation Dimensions

The rectangular sheet has dimensions:

  • Length (\(L\)) = 48 cm
  • Width (\(W\)) = 36 cm

Squares of side length \(x\) cm are cut from each corner. This results in an open box with:

  • Height (\(H_{box}\)) = \(x\) cm
  • Length of base (\(L_{box}\)) = \(48 - 2x\) cm
  • Width of base (\(W_{box}\)) = \(36 - 2x\) cm

For the dimensions to be physically possible, \(x\) must be positive and less than half the smallest dimension of the sheet. Therefore, \(0 < x < \frac{36}{2}\), which means \(0 < x < 18\) cm.

Calculating Box Volume

The volume (\(V_{box}\)) of the box is calculated as the product of its length, width, and height:

\(V_{box} = L_{box} \times W_{box} \times H_{box}\)

The given volume is 5.12 litres. We need to convert this to cubic centimeters (cm\(^3\)) since the sheet dimensions are in cm:

\(V_{box} = 5.12 \text{ litres} \times \frac{1000 \text{ cm}^3}{1 \text{ litre}} = 5120 \text{ cm}^3\)

Solving the Volume Equation

Substitute the dimensions into the volume formula:

\((48 - 2x)(36 - 2x)(x) = 5120\)

Simplify the equation by factoring out 2 from the length and width terms:

\([2(24 - x)][2(18 - x)](x) = 5120\) \(4x(24 - x)(18 - x) = 5120\)

Divide both sides by 4:

\(x(24 - x)(18 - x) = \frac{5120}{4}\) \(x(24 - x)(18 - x) = 1280\)

Expanding this expression leads to a cubic equation:

\(x(432 - 24x - 18x + x^2) = 1280\) \(x(x^2 - 42x + 432) = 1280\) \(x^3 - 42x^2 + 432x - 1280 = 0\)

Evaluating Options for x

Given the cubic nature of the equation, testing the provided options is an efficient strategy for exam questions.

Checking x = 8 cm

Let's substitute \(x = 8\) cm into the simplified equation \(x(24 - x)(18 - x) = 1280\):

\(8 \times (24 - 8) \times (18 - 8)\) \(= 8 \times 16 \times 10\) \(= 8 \times 160\) \(= 1280\)

Since the calculation yields 1280, which matches the required volume in cm\(^3\), \(x = 8\) cm is the correct solution.

Final Answer

The value of \(x\) is 8 cm.

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Important Questions from 3-D Mensuration

  1. Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]

  2. Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?

  3. The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
  4. The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
    (Use $\pi = \frac{22}{7}$)

  5. A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
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