A solid wooden pillar shaped as a right circular cylinder has a radius of 5 cm and a height of 12 cm. A coaxial cylindrical core with a radius of 3 cm is hollowed out completely through its entire height. What percentage of the initial volume of the wooden pillar was extracted?

The correct option is 36%.

Step 1: Understand the Given Dimensions
We are given a solid right circular cylinder with:
• Initial radius of the pillar, $R=5\text{cm}$
• Total height of the pillar, $h=12\text{cm}$
• Radius of the hollowed-out cylindrical core, $r=3\text{cm}$

Step 2: Formula for the Volume of a Cylinder
The volume of a cylinder is given by the formula:
$V=\pi r^{2}h$

Step 3: Calculate the Initial Volume of the Wooden Pillar
Using the initial radius $R=5\text{cm}$ , the initial volume $V_{\text{initial}}$ is:
$V_{\text{initial}}=\pi \times 5^2\times 12$
$V_{\text{initial}}=\pi \times 25\times 12=300\pi {\text{cm}}^3$

Step 4: Calculate the Volume of the Extracted Core
The hollowed-out core has a radius of $r=3\text{cm}$ and extends through the entire height $h=12\text{cm}$ . The extracted volume $V_{\text{extracted}}$ is:
$V_{\text{extracted}}=\pi \times 3^2\times 12$
$V_{\text{extracted}}=\pi \times 9\times 12=108\pi {\text{cm}}^3$

Step 5: Calculate the Percentage of Volume Extracted
The percentage of the initial volume extracted is given by:
$\text{Percentage Extracted}=\left(\frac{V_{\text{extracted}}}{V_{\text{initial}}}\right)\times 100\%$
$\text{Percentage Extracted}=\left(\frac{108\pi }{300\pi }\right)\times 100\%$
$\text{Percentage Extracted}=\frac{108}{300}\times 100\%=\frac{108}{3}\%=36\%$

Thus, 36% of the initial volume of the wooden pillar was extracted.