Question

a solid oblique cone with a slant length of 17 units is placed inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. what is the unfilled volume inside the cylinder? 320π cubic units 597π cubic units 640π cubic units 725π cubic units

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \). Given \( r = 8 \) and \( h = 15 \), we substitute these values:
\( V_{cylinder} = \pi \times 8^2 \times 15 = \pi \times 64 \times 15 = 960\pi \)

Step2: Calculate the volume of the cone

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \). The radius of the cone is the same as the cylinder (\( r = 8 \)) and the height of the cone is the same as the cylinder (\( h = 15 \)) (since the base is congruent and it's placed inside the cylinder). Substituting the values:
\( V_{cone} = \frac{1}{3}\pi \times 8^2 \times 15 = \frac{1}{3}\pi \times 64 \times 15 = 320\pi \)

Step3: Calculate the unfilled volume

The unfilled volume is the volume of the cylinder minus the volume of the cone:
\( V_{unfilled} = V_{cylinder} - V_{cone} = 960\pi - 320\pi = 640\pi \)

Answer:

\( 640\pi \) cubic units (corresponding to the option "640π cubic units")