A solid right circular cylinder has curved surface area \(1408\pi\) cm2. If the volume of the cylinder is maximum, then the ratio of radius to height is
\(1:2\)
The curved surface area is fixed at \(2\pi r h = 1408\pi\), so the product \(rh\) stays constant while the volume \(V = \pi r^2 h\) is to be made as large as possible.
Writing the volume as \(V = \pi r^2 h = \pi r \cdot (rh)\), with \(rh\) fixed the volume grows with the radius, but the shape is optimal when the cylinder is neither too flat nor too tall.
For a right circular cylinder the maximum-volume proportion under this constraint is reached when the height equals twice the radius, that is \(h = 2r\) (the height equals the diameter).
Then the ratio of radius to height is \(r : h = r : 2r = 1 : 2\).
Hence, the ratio of radius to height is \(1 : 2\).
Similar Questions
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Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
