Question : The sum of the length, breadth, and height of a cuboid box is 20 cm and the total surface area of a cuboid is 256 sq. cm. What is the maximum length (in approximate) of a stick that can be placed inside the cuboid box?
Option 1: 16 cm
Option 2: 24 cm
Option 3: 32 cm
Option 4: 12 cm
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Correct Answer: 12 cm
Solution :
The sum of the length, breadth, and height of a cuboid box is 20 cm.
⇒ $l + b + h = 20$, Where $l$ = length, $b$ = breadth, $h$ = height
The total surface area of the cuboid = 256 sq. cm.
⇒ $2(lb + bh + lh) = 256$
⇒ $lb+bh+lh=128$
The maximum length of a stick that can be placed inside the cuboid is the diagonal of the cuboid and
Diagonal of cuboid = $\sqrt{l^2 + b^2 + h^2 }$
Using formula (a + b + c)
2
= a
2
+ b
2
+ c
2
+ 2(ab + bc + ca), we get,
$20^2 = l^2 + b^2 + h^2+ 256$
⇒ $400 - 256 = l^2 + b^2 + h^2$
⇒ $l^2 + b^2 + h^2 = 144$
⇒ $ \sqrt{l^2 + b^2 + h^2} = \sqrt{144}$
$\therefore\sqrt{l^2 + b^2 + h^2}= 12$
Hence, the correct answer is 12 cm.
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