Theory:

1. A trash bin is in the shape of a cylinder of diameter \(28 \ cm\) and height \(40 \ cm\). Find the cost of painting the trash bin (including lid) at \(₹ 5\) per \(cm^2\). 
 
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Solution:
 
Diameter of the base \((d)\) \(=\) \(28 \ cm\)
 
Radius of the base \((r)\) \(=\) d2=282=14 \(cm\)
 
Height of the trash bin \((h)\) \(=\) \(40 \ cm\)
 
Total surface area of the right circular cylinder \(=\) \(2 \pi r (r + h)\) sq. units
 
\(=\) \(2 \times \frac{22}{7} \times 14 (14 + 40)\)
 
\(=\) \(2 \times 22 \times 2 \times 54\)
 
\(=\) \(4752\)
 
The total surface area of the trash bin \(=\) \(4752\) \(cm^2\)
 
Cost of painting the trash bin per \(cm^2\) \(=\) \(₹5\)
 
Cost of painting the trash bin for \(4752\) \(cm^2\):
 
\(=\) \(4752 \times 5\)
 
\(=\) \(23760\)
 
Therefore, the cost of painting the trash bin is \(₹ 23760\).
 
 
2. The hollow cylinder height \(8.4 \ cm\) has the internal and external radii of \(2 \ cm\) and \(5 \ cm\), respectively. Find the curved surface area of the hollow cylinder.
 
Solution:
 
Height of the cylinder \(=\) \(8.4 \ cm\)
 
Internal radius, \(r\) \(=\) \(2 \ cm\)
 
External radius, \(R\) \(=\) \(5 \ cm\)
 
Curved surface area of a hollow cylinder \(=\) \(2 \pi (R + r)h\) sq. units
 
\(=\) \(2 \times \frac{22}{7} \times (5 + 2) \times 8.4\)
 
\(=\) \(2 \times \frac{22}{7} \times 7 \times 8.4\)
 
\(=\) \(2 \times 22 \times 8.4\)
 
\(=\) \(369.6\)
 
Therefore, the curved surface area of the hollow cylinder is \(369.6\) \(cm^2\).
 
Important!
The value of \(\pi\) should be taken as \(\frac{22}{7}\) unless its value is shared in the problem.