Question

The internal and external diameters of a hollow hemispherical vessel are 24cm and 25cm respectively. If the cost of painting $1c{m}^2$ surface area is$Rs.5.25$, find the total cost of painting the vessel all over. $\left(use\mathrm{\pi }=\frac{22}{7}\right)$

Answer 1

Step 1: Given information

The internal diameter of a hollow hemispherical vessel,$d=24cm$.

Internal radius of the hollow hemispherical vessel $=r=\frac{24}{2}=12cm.$

External diameter of the hollow hemispherical vessel,$D=25cm.$

The external radius of a hemispherical vessel. $R=\frac{25}{2}=12.5cm.$

Step 2: The required formulas

Area of the outer surface$=2{\mathrm{\pi R}}^2.$

Area of the inner surface$=2{\mathrm{\pi r}}^2.$

Step 3: Calculate the total cost of painting the vessel

Total surface area to be painted
= External curved surface + Internal curved surface + Area of ring

$$\begin{gathered} =2\pi R^2+2\pi r^2+\pi \left(R^2-r^2\right) \\ =\pi \left(2R^2+2r^2+R^2-r^2\right) \\ =\frac{22}{7}\times \left(3R^2+r^2\right) \\ =\frac{22}{7}\times \left(3\times {\left(12.5\right)}^2+{\left(12\right)}^2\right) \\ =\frac{22}{7}\times \left(468.75+144\right) \\ =\frac{22}{7}\times 612.75 \\ =1925.78sq.cm \end{gathered}$$

The rate of painting = $5.25$paise per sq. cm.

Hence, The total cost of painting=Area × Rate=$1925.78\times 5.25=Rs.10110.38$