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The maximum length of a pencil that can be kept in a rectangular box of dimensions 8 cm × 6 cm × 2 cm, is:2√13 cm2√14 cm2√26 cm10√2 cm

Question

The maximum length of a pencil that can be kept in a rectangular box of dimensions 8 cm × 6 cm × 2 cm, is:

  • 2√13 cm
  • 2√14 cm
  • 2√26 cm
  • 10√2 cm
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Solution

1. Break Down the Problem

To find the maximum length of a pencil that can fit inside a box, we need to calculate the space diagonal of the rectangular box. The dimensions of the box are given as:

  • Length (l) = 8 cm
  • Width (w) = 6 cm
  • Height (h) = 2 cm

2. Relevant Concepts

The formula for the diagonal dd of a rectangular box with dimensions ll, ww, and hh is given by: d=l2+w2+h2 d = \sqrt{l^2 + w^2 + h^2}

3. Analysis and Detail

Now, we will calculate the diagonal using the given dimensions:

  1. Calculate l2l^2: l2=82=64 l^2 = 8^2 = 64
  2. Calculate w2w^2: w2=62=36 w^2 = 6^2 = 36
  3. Calculate h2h^2: h2=22=4 h^2 = 2^2 = 4
  4. Sum the squares: l2+w2+h2=64+36+4=104 l^2 + w^2 + h^2 = 64 + 36 + 4 = 104
  5. Take the square root to find the diagonal: d=104=4×26=226 d = \sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26}

4. Verify and Summarize

We confirmed all calculations are accurate. Therefore, the maximum length of the pencil that can be kept in the box is 226 cm2\sqrt{26} \text{ cm}.

Final Answer

The maximum length of a pencil that can be kept in the rectangular box is 226 cm2\sqrt{26} \text{ cm}.

This problem has been solved

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