Final answer:
To find the volume of the remaining solid, subtract the volume of the conical cavity from the volume of the original solid cylinder. The volume of the remaining solid is 628 cm³. To find the total surface area of the remaining solid, subtract the curved surface area of the conical cavity from the curved surface area of the cylinder. The total surface area of the remaining solid is 171.94 cm².
Step-by-step explanation:
To find the volume of the solid remaining after hollowing out the conical cavity, we need to find the volume of the original solid cylinder and subtract the volume of the conical cavity.
1. Volume of the cylinder: V = πr²h = 3.142 x (5 cm)² x 12 cm = 942 cm³
2. Volume of the conical cavity: V = (1/3)πr²h = (1/3) x 3.142 x (5 cm)² x 12 cm = 314 cm³
3. Volume of the remaining solid: V remaining = V cylinder - V conical cavity = 942 cm³ - 314 cm³ = 628 cm³
To find the total surface area of the remaining solid, we need to consider the curved surface area of the cylinder and the curved surface area of the conical cavity that was hollowed out.
4. Curved surface area of the cylinder: A = 2πrh = 2 x 3.142 x 5 cm x 12 cm = 376.8 cm²
5. Curved surface area of the conical cavity: A = πrl, where r is the radius of the conical cavity and l is the slant height. Since the diameter of the cylinder and conical cavity are the same, their radii are the same. The slant height of the conical cavity can be found using the Pythagorean theorem with the radius (5 cm) and the height of the conical cavity (12 cm): l = sqrt(r² + h²) = sqrt(5² + 12²) = 13 cm
Now we can calculate the curved surface area of the conical cavity: A = 3.142 x 5 cm x 13 cm = 204.86 cm²
6. Total surface area of the remaining solid: A total = A cylinder - A conical cavity = 376.8 cm² - 204.86 cm² = 171.94 cm²