Question:
A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is \(\frac{5}{4}\) times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is
A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is \(\frac{5}{4}\) times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is
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Express material in terms of one variable using volume constraint.
Updated On: Mar 23, 2026
- \(\dfrac{2}{3}\)
- \(\dfrac{1}{2}\)
- \(\dfrac{4}{5}\)
- (1)/(3)
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The Correct Option is C
Solution and Explanation
Material used:
\[ M = 2 \pi r h + \frac{5}{4} \pi r^2 \]
For fixed volume \(V = \pi r^2 h\),
\[ M = \frac{2V}{r} + \frac{5}{4} \pi r^2 \]
Minimizing gives \(h = \frac{5}{4} r \implies \frac{r}{h} = \frac{4}{5}\).
\[ M = 2 \pi r h + \frac{5}{4} \pi r^2 \]
For fixed volume \(V = \pi r^2 h\),
\[ M = \frac{2V}{r} + \frac{5}{4} \pi r^2 \]
Minimizing gives \(h = \frac{5}{4} r \implies \frac{r}{h} = \frac{4}{5}\).
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