Mathematics
A hollow square-shaped tube open at both ends is made of iron. The internal square is of 5 cm side and the length of the tube is 8 cm. There are 192 cm3 of iron in this tube. Find its thickness.
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Answer
Given:
Volume of iron = 192 cm3
Length of the tube = 8 cm
Internal side of the square = 5 cm
Let x be the thickness of the tube.
External side of the square = (5 + 2x) cm
Internal volume of the tube = Internal Side x Internal Side x length
= 5 x 5 x 8 cm3
= 200 cm3
External volume of the tube = External side x External side x length
= (5 + 2x) x (5 + 2x) x 8 cm3
= (4x2 + 20x + 25) x 8 cm3
= 32x2 + 160x + 200 cm3
Volume of iron = External volume - Internal volume
⇒ (32x2 + 160x + 200) - 200 cm3 = 192 cm3
⇒ 32x2 + 160x + 200 - 200 - 192 = 0
⇒ 32x2 + 160x - 192 = 0
Divide through by 32:
⇒ x2 + 5x - 6 = 0
⇒ x2 + 6x - 1x - 6 = 0
⇒ x(x + 6) - 1(x + 6) = 0
⇒ (x + 6)(x - 1) = 0
⇒ x = - 6 or 1
Since thickness cannot be negative, x = 1.
Hence, the thickness of the pipe is 1 cm.
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