Question

A narrow box is 6√2cm long and 4√2 cm wide Find its height if the length of the diagonal is 4√7cm​

Answer 1

$\sf Given \begin{cases} & \sf{Length\:of\:box\:,l = \bf{6\sqrt{2}\:cm}} \\ & \sf{Breadth\:of\:box\:,b = \bf{4\sqrt{2}\:cm}} \\ & \sf{Length\:of\: diagonal\:of\:box\:,d = \bf{4\sqrt{7}\:cm}} \end{cases}$

To find: Height of box?

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$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

⋆ Formula to find diagonal of a cuboid is given by,

$\star\;{\boxed{\sf{\pink{Diagonal_{\;(cuboid)} = \sqrt{(length)^2 + (breadth)^2 + (height)^2}}}}}$

$:\implies\sf d = \sqrt{l^2 + b^2 + h^2}$

Here,

• l = 6√2 cm
• b = 4√2 cm
• d = 4√7 cm

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$\dag\;{\underline{\frak{Now,\:Putting\:values\;in\;formula,}}}$

$:\implies\sf 4\sqrt{7} = \sqrt{(6\sqrt{2})^2 + (4\sqrt{2})^2 + h^2}\\ \\ \\ :\implies\sf 4\sqrt{7} = \sqrt{72 + 32 + h^2}$

$:\implies\sf (4\sqrt{7})^2 = \bigg(\sqrt{72 + 32 + h^2}\bigg)^2\qquad\bigg\lgroup\bf Squaring\:both\:sides\bigg\rgroup$

$:\implies\sf (4\sqrt{7})^2 = 72 + 32 + h^2\\ \\ \\ :\implies\sf 112 = 104 + h^2\\ \\ \\ :\implies\sf h^2 = 112 - 104\\ \\ \\ :\implies\sf h^2 = 8\\ \\ \\ :\implies\sf \sqrt{h^2} = \sqrt{8}\\ \\ \\ :\implies\sf h = \sqrt{8}\\ \\ \\:\implies{\underline{\boxed{\frak{\purple{h = 2 \sqrt{2}\:cm}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Height\:of\:narrow\:box\:is\: \bf{2\sqrt{2}\:cm}.}}}$

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$\qquad\boxed{\underline{\underline{\pink{\bigstar \: \bf\:Formulas\:related\:to\;cuboid\:\bigstar}}}}$

• $\sf Area\:of\:base\:of\:cuboid = \bf{length \times breadth}$

• $\sf Total\:surface\:area\:of\;cuboid = \bf{2(lb + bh + hl)}$

• $\sf Curved\:surface\:area\:of\:cuboid = \bf{2(l + b) \times h}$

Answer 2

Answer:

★ǫᴜᴇsᴛɪᴏɴ★

• A narrow box is 6√2cm long and 4√2 cm wide. Find its height if the length of the diagonal is 4√7cm.

★ANSWER★

Given :-

• Length of box = 6√2 cm
• Breadth of box = 4√2 cm
• Diagonal of box = 4√7 cm

To find :-

• Height of the box

Formula used :-

Diagonal of a cuboid is given by

$\small \bf d \: = \sqrt{ {l}^{2} + {b}^{2} + {h}^{2} }$

where, symbol have the meanings

• l = Length of Cuboid

• b = Breadth of Cuboid

• h = Height of Cuboid

• d = Diagonal of Cuboid

Solution :-

★ According to statement

• Length of box = 6√2 cm

• Breadth of box = 4√2 cm

• Diagonal of box = 4√7 cm

Let Height of the box be 'h' cm.

Using the formula, the diagonal (d) of box is

$\small \bf d \: = \sqrt{ {l}^{2} + {b}^{2} + {h}^{2} }$

Substitute the values of d, l and b.

$\small \bf 4 \sqrt{7} \: = \sqrt{ {(6 \sqrt{2}) }^{2} + {(4 \sqrt{2}) }^{2} + {h}^{2} }$

$\bf \implies \:4 \sqrt{7} = \sqrt{72 \ + 32 + {h}^{2} }$

$\small \bf squaring \: both \: sides$

$\bf \implies \: 112 = 104 + {h}^{2}$

$\bf \implies \: {h}^{2} = 112 - 104$

$\bf \implies \: {h}^{2} = 8$

$\bf \implies \:h = \sqrt{8} = 2 \sqrt{2} \: cm$

$\bf \implies \:height \: of \: box \: = 2 \sqrt{2} \: cm$

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