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
- 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).
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 $\pi = \frac{22}{7}$)- 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:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
The length and breadth of a cuboid are 50 cm and 75 cm, respectively. If the length of the diagonal is $17\sqrt{29}$ cm, then what is the volume (in $cm^{3}$) of the cuboid?
