Question

0-24 An ant walks along a helical path on a solid cylindrical object of diameter 5 cm and height 5 cm. It starts walking from point X goes around the object, and reaches point Y, which is vertically below point X as shown in the figure, How much distance did the ant walk?
5 cm
A √(25 + 257π²) cm
B. √(25 + 100π²) cm
C. 25 + 25π² cm
D. 5 + 5π cm​

Answer 1

Join X and Y, cut the outer cover of the cylindrical object and spread the cover out (left side of XY line towards right). It looks like,

$\setlength{\unitlength}{0.6mm}\begin{picture}(5,5)\thicklines\framebox(157.08,50){}\multiput(-163,50)(166,0){2}{ X }\multiput(-163,-3)(166,0){2}{ Y }\qbezier(-157.08,50)(-78.54,25)(0,0)\multiput(-137.445,43.75)(19.635,-6.25){7}{\vector(3,-1){0}}\multiput(-85,-7)(0,60){2}{ 2\pi r }\multiput(3,25)(-167,0){2}{ h }\end{picture}$

It looks like a rectangle of length $2\pi r$ and breadth $h,$ where $r$ is radius and $h$ is height of the cylinder. The diagonal shown is the path traced by the ant.

So the total distance walked by the ant, or the length of the diagonal is,

$\longrightarrow d=\sqrt{(2\pi r)^2+h^2}$

$\longrightarrow d=\sqrt{4\pi^2r^2+h^2}$

In the question,

• $r=\dfrac{5}{2}\ cm$

• $h=5\ cm$

Then,

$\longrightarrow d=\sqrt{4\pi^2\left(\dfrac{5}{2}\right)^2+5^2}$

$\longrightarrow\underline{\underline{d=\sqrt{25+25\pi^2}\ cm}}$

$\longrightarrow\underline{\underline{d=5\sqrt{1+\pi^2}\ cm}}$

Answer 2

$\huge\bf\fbox\red{Answer:-}$

The total distance walked by the ant, or the length of the diagonal is,

$\longrightarrow d=\sqrt{(2\pi r)^2+h^2}$

$\longrightarrow d=\sqrt{4\pi^2r^2+h^2}$

In the question,

$r=\dfrac{5}{2}\ cm$

$h=5\: cm$

Then,

$\longrightarrow d=\sqrt{4\pi^2\left(\dfrac{5}{2}\right)^2+5^2}$

$\longrightarrow\underline{\underline{d=\sqrt{25+25\pi^2}\ cm}}$

$\longrightarrow\underline{\underline{d=5\sqrt{1+\pi^2}\ cm}}$