From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of the remaining solid.
\(343-21\pi\;\text{cm}^3\) (≈ 277.0 cm³)
Step 1: Write down what is given.
Step 2: Find the volume of the cube.
Formula: \( V_{\text{cube}} = (\text{side})^3 \)
\( V_{\text{cube}} = 7^3 = 343\,\text{cm}^3 \)
Step 3: Find the volume of the cone (the hollowed part).
Formula: \( V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h \)
\( V_{\text{cone}} = \tfrac{1}{3}\pi (3^2)(7) \)
\( V_{\text{cone}} = \tfrac{1}{3}\pi (9)(7) = 21\pi\,\text{cm}^3 \)
Step 4: Subtract the cone volume from the cube volume.
Remaining volume = \( 343 - 21\pi \;\text{cm}^3 \)
Step 5: Approximate using \(\pi ≈ 3.1416\).
\( 21\pi ≈ 21 × 3.1416 ≈ 65.97 \)
Remaining volume = \( 343 - 65.97 ≈ 277.0\,\text{cm}^3 \)
Final Answer: The volume of the remaining solid is about 277 cm³.