# CAT 2003 | Question: 2-65

67 views

Consider a cylinder of height $h$ cms and radius $r=\frac{2}{\pi}$ cms as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of $n$ turns (in other words, the string‟s length is the minimum length required to wind $n$ turns.)

The same string, when wound on the exterior four walls of a cube of side $n$ cms, starting at point C and ending at point D, can give exactly one turn (see figure, not drawn to scale).

How is $h$ related to $n?$

1. $h=\sqrt{2}n$
2. $h=\sqrt{17}n$
3. $h=n$
4. $h=\sqrt{13}n$

edited

1 vote

Unfold the cylinder along the line AB,

It’ll form a rectangle with width $=2\pi r = 2 \pi \times \frac{2}{\pi} = 4$ cms, and length$=h$ cms.

length of $1$ roll of string $=\sqrt{(\frac{h}{n})^2 + 4^2} = \frac{1}{n} \sqrt{h^2 + 16n^2}$

length of string $=n \times \frac{1}{n}\sqrt{h^2 + 16n^2} = \sqrt{h^2 + 16n^2}$

Unfold the cube, then the length of the string is nothing but the hypotenuse of rectangle with height=$n$ and width=$4n$.

length of string $= \sqrt{n^2 + (4n)^2} = \sqrt{17 n^2}$

So, $\sqrt{17 n^2} = \sqrt{h^2 + 16n^2} \implies h=n$
626 points 1 2 14
selected by

## Related questions

1
49 views
Consider a cylinder of height $h$ cms and radius $r=\frac{2}{\pi}$ cms as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of $n$ turns (in other words, the string ... see figure, not drawn to scale). The length of the string, in cms, is $\sqrt{2}n$ $\sqrt{12}n$ $n$ $\sqrt{13}n$
2
391 views
Consider a cylinder of height h cms and radius $r=\frac{2}{\pi}$ cms as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of n turns (in other words, ... between two consecutive turns? $\frac{h}{n}$ $\frac{h}{\sqrt{n}}$ $\frac{h}{n^2}$ Cannot be determined with given information
There are $8436$ steel balls, each with a radius of $1$ centimeter, stacked in a pile, with $1$ ball on top, $3$ balls in second layer, $6$ in the third layer, $10$ in the fourth, and so on. The number of horizontal layers in the pile is $34$ $38$ $36$ $32$
Let A and B two solid spheres such that the surface area of B is $300\%$ higher than the surface area of A. The volume of area is found to be $k\%$ lower than the volume of B. The value of $k$ must be ________ $85.5$ $92.5$ $90.5$ $87.5$
$70\%$ of the employees in a multinational corporation have VCD players, $75$ percent have microwave ovens, $80$ percent have ACs and $85$ percent have washing machines. At least what percentage of employees has all four gadgets? $15\%$ $5\%$ $10\%$ Cannot be determined