Chapter 12.3, Problem 24HP

To determine

Tocheck:which of the given cylinder has greater volume when both the sheets are equal sized.

Answer to Problem 24HP

volume of the cylinder along with the length is greater.

Explanation of Solution

Given:

Two equal-sized sheets of paper are rolled along with the length and along the width as shown in given figure.

  

Calculation:

According to the given figure in the question, there is one cuboid and two cylinder having equal dimensions.

Now, let the length of the paper be x and width of the paper be y, where $x>y$ .

  $\begin{array}{l}\therefore 2\pi r=x\\ \Rightarrow r=\frac{x}{2\pi }\end{array}$

And

  $\begin{array}{l}\therefore 2\pi r=y\\ \Rightarrow r=\frac{y}{2\pi }\end{array}$

Also, it is known that volume of the cylinder is $\pi r^{2}h$ .

Now, volume of the cylinder along with the length is

  $\begin{array}{l}V=\pi r^{2}h\\ =\pi {\left(\frac{x}{2\pi }\right)}^{2}y=\frac{x^{2}y}{2}\end{array}$

And, volume of the cylinder along with the widthis

  $\begin{array}{l}V=\pi r^{2}h\\ =\pi {\left(\frac{y}{2\pi }\right)}^{2}y=\frac{y^{2}x}{2}\end{array}$

As $x>y$ ,

Therefore, volume of the cylinder along with the length is greater than volume of the cylinder along with the width.

Hence, volume of the cylinder along with the length is greater.