### 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 $$cm^2$$.

Solution:

Diameter of the base $$(d)$$ $$=$$ $$28 \ cm$$

Radius of the base $$(r)$$ $$=$$ $\frac{d}{2}=\frac{28}{2}=14$ $$cm$$

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.