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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