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प्रश्न
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
विकल्प
12 cm
6 cm
18 cm
3 cm
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उत्तर
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is 12 cm.
Explanation:
Let the length, breadth and height of a rectangular block be `a/r`, a abd ar. [Since they are is G.P]
∴ Volume = l × b × h
216 = `a/r xx a xx ar`
⇒ a3 = 216
⇒ a = 6
Now total surface area = `2[lb + bh + lh]`
252 = `2[a/r * a + a * ar + a/r]`
⇒ 252 = `2[a^2/r + a^2r + a^2]`
⇒ 252 = `2a^2 [1/r + r + 1]`
⇒ 252 = `2 xx (6)^2 [(1 + r^2 + r)/r]`
⇒ 252 = `72[(1 + r^2 + r)/r]`
⇒ `252/72 = (1 + r + r^2)/r`
⇒ `7/2 = (1 + r + r^2)/r`
⇒ 2 + 2r + 2r2 = 7r
⇒ 2r2 – 5r + 2 = 0
⇒ 2r2 – 4r – r + 2 = 0
⇒ 2r(r – 2) –1(r – 2) = 0
⇒ (r – 2)(2r – 1) = 0
⇒ r – 2 = 0 and 2r – 1 = 0
∴ r = 2, `1/2`
Therefore, the three edge are:
If r = 2 then edges are 3, 6, 12
If r = `1/2` then edges are 12, 6, 3
So, the length of the longest edge = 12
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