A solid metallic cylinder with a radius of 6 cm is modified by boring five uniform cylindrical channels, each having a radius of 1.5 cm, completely along its full height. Determine the approximate percentage of the original cylinder's volume that was extracted during this procedure.

The correct option is 31.25%.

Step-by-Step Explanation:

Step 1: Understand the Geometry of the Cylinder

Let the original solid cylinder have a radius $R$ and height $h$.

From the problem statement:

Radius of the original cylinder, $R=6\text{cm}$

Height of the original cylinder = $h$

The formula for the volume of a cylinder is given by:

$V=\pi r^{2}h$

Therefore, the original volume ($V_{\text{original}}$) of the solid cylinder is:

$V_{\text{original}}=\pi R^{2}h=\pi \times 6^2\times h=36\pi h$

Step 2: Calculate the Volume Extracted

Five uniform cylindrical channels are bored through the cylinder along its full height $h$.

Radius of each bored channel, $r=1.5\text{cm}$

The volume of one cylindrical channel is:

$V_{\text{channel}}=\pi r^{2}h=\pi \times {\left(1.5\right)}^2\times h=2.25\pi h$

Since there are 5 identical channels bored out, the total extracted volume ($V_{\text{extracted}}$) is:

$V_{\text{extracted}}=5\times V_{\text{channel}}=5\times 2.25\pi h=11.25\pi h$

Step 3: Calculate the Percentage of Volume Extracted

The percentage of the original volume that was extracted is given by:

$\text{Percentage Extracted}=\left(\frac{V_{\text{extracted}}}{V_{\text{original}}}\right)\times 100$

Substituting the calculated volumes into the formula:

$\text{Percentage Extracted}=\left(\frac{11.25\pi h}{36\pi h}\right)\times 100$

Simplifying the fraction by canceling out $\pi h$:

$\text{Percentage Extracted}=\frac{11.25}{36}\times 100=0.3125\times 100=31.25\%$

Hence, approximately 31.25% of the original cylinder's volume was extracted.