Question : The sum of the length, breadth, and height of a cuboid is 20 cm. If the length of the diagonal is 12 cm, then find the total surface area of the cuboid.
Option 1: 364 cm2
Option 2: 256 cm2
Option 3: 356 cm2
Option 4: 264 cm2
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Correct Answer: 256 cm 2
Solution :
Given: The sum of the length, breadth, and height of a cuboid is 20 cm.
The length of the diagonal is 12 cm.
Use the formulas,
The length of the diagonal of the cuboid = $\sqrt{l^2+b^2+h^2}$,
The total surface area of the cuboid = $2(lb+bh+lh)$,
where $l$, $b$, and $h$ are the length, breadth, and height respectively.
According to the question,
$\sqrt{l^2+b^2+h^2}=12$
⇒ $l^2+b^2+h^2=144$ (equation 1)
Also, $(l+b+h)=20$ (equation 2)
Squaring both sides of the equation 2,
$(l+b+h)^2=(20)^2$
⇒ $l^2+b^2+h^2+2(lb+bh+lh)=400$
Substitute the value of $l^2+b^2+h^2$ from the equation 1,
⇒ $144+2(lb+bh+lh)=400$
⇒ $2(lb+bh+lh)=400–144$
⇒ $2(lb+bh+lh)=256$ cm
2
Hence, the correct answer is 256 cm
2
.
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