A cylindrical cavity with a radius of 3 cm and a depth of 2 cm is created inside one end of a solid cylinder with a radius of 5 cm and a height of 7 cm. Find the total surface area (in cm²) of the resulting solid.

Correct option is D

Given:

Radius of the solid cylinder, R = 5 cm
Height of the solid cylinder, h = 7 cm
Radius of the cylindrical cavity, r = 3 cm
Depth of the cylindrical cavity, d = 2 cm

Concept Used:

The total surface area of the resulting solid will consist of:
The curved surface area of the outer solid cylinder.
The circular area of the base of the solid cylinder.
The curved surface area of the cylindrical cavity (which is now inside the cylinder).
The circular area of the opening of the cavity (which is exposed).

Formula Used:

Curved surface area of a cylinder: $$A_{curved} = 2 × π × r × h$$​​
Area of the base of the cylinder: $$A_{base} = π × r²$$​​
Total surface area of the resulting solid: $$A_{total }= 2 × π × R × h + π × R² + 2 × π × r × d + π × r²$$​​

Solution:

Curved surface area of the outer cylinder:
​$$A_{curved_\ outer }= 2 × π × R × h = 2 × π × 5 × 7 = 70 × π cm²$$​​
Both Base area of the solid cylinder:
​$$A_{base} = 2π × R² = π × 5² = 50× π cm²$$​​
Curved surface area of the cylindrical cavity:
​$$A_{curved_\ cavity} = 2 × π × r × d = 2 × π × 3 × 2 = 12 × π cm²$$​​
Area of the Base of the cavity:
​$$A_{opening} = π × r² = π × 3² = 9 × π cm²$$ 

Area of the top of the cavity which is opened = $$A_{opening} = π × r² = π × 3² = 9 × π cm²$$​​
​​Total surface area of the resulting solid:
​$$A_{total} = 70 × π + 50 × π + 12 × π + 9 × π - 9 × π= 132 × π cm²$$​​

The total surface area of the resulting solid is 132 × π cm².