Question

By taking out a hemisphere from both the ends of a wooden solid cylinder, an item is formed. If the height of the cylinder is 10 cm and radius of the base is 3.5 cm, find the total surface area of the item.

Hint:

Always exclude the base area when hemispheres are removed from a cylinder. Add the curved surface area of the cylinder and two hemispheres only.

Answer 1

Step 1: Given data. 
Radius, \( r = 3.5 \, \text{cm} \) 
Height of the cylinder, \( h = 10 \, \text{cm} \) 
Step 2: Total surface area (TSA) of the item. 
Since hemispheres are taken out from both ends, there are no circular bases. Thus, \[ \text{TSA} = \text{Curved surface area of cylinder} + 2 \times \text{Curved surface area of hemisphere} \] Step 3: Write the formulas. 
\[ \text{CSA of cylinder} = 2\pi rh \] \[ \text{CSA of one hemisphere} = 2\pi r^2 \] Therefore, \[ \text{TSA} = 2\pi rh + 2(2\pi r^2) = 2\pi r(h + 2r) \] Step 4: Substitute the given values. 
\[ \text{TSA} = 2 \times 3.14 \times 3.5 (10 + 2 \times 3.5) \] \[ = 6.28 \times 3.5 \times 17 = 6.28 \times 59.5 = 373.66 \, \text{cm}^2 \] Step 5: Conclusion. 
Hence, the total surface area of the item is approximately 373.66 cm$^2$.