Question

The front compound wall of a house is decorated by wooden spheres of diameter $ 21 \mathrm{cm}$, placed on small supports as shown in fig. Eight such spheres are used forth is purpose, and are to be painted silver. Each support is a cylinder of radius $ 1.5 \mathrm{cm}$ and height $ 7 \mathrm{cm}$ and is to be painted black. Find the cost of paint required if silver paint costs $ 25 \mathrm{paise} \mathrm{per} {\mathrm{cm}}^{2}$ and black paint costs $ 5 \mathrm{paise} \mathrm{per} {\mathrm{cm}}^{2}$.

Answer 1

Step 1: Determination of total area to be painted with silver paint.

Diameter of a wooden sphere $=21\mathrm{cm}$

The radius of a wooden sphere,$\mathrm{r}=\left(\frac{21}{2}\right)\mathrm{cm}=10.5\mathrm{cm}$

• The surface area of the wooden sphere $=4{\mathrm{\pi r}}^2$
  $$\begin{gathered} =4\times \frac{22}{7}\times 10.5^2 \\ =1386\mathrm{cm}2 \end{gathered}$$
• The area of the circular end of the cylindrical support$={\mathrm{\pi r}}^2$
  $$\begin{gathered} =\frac{22}{7}\times 1.5^2 \\ =7.07{\mathrm{cm}}^2 \end{gathered}$$
• Area to be painted silver $=[8\times (1386-7.07\left)\right]$
  $$\begin{gathered} =8\times 1378.93 \\ =11031.44{\mathrm{cm}}^2 \end{gathered}$$

Step 2: Determination of silver paint cost

• Cost for painting $1{\mathrm{cm}}^2$ with silver color $\mathrm{Rs}0.25$
  Thus, the cost of painting $11031.44{\mathrm{cm}}^2$ with silver color $=\mathrm{Rs}(11031.44\times 0.25)$ $=\mathrm{Rs}2757.86$

Step 3: Determination of total area to be painted with black paint.

• The radius of the circular end of a cylindrical support $\left(\mathrm{r}\right)=1.5\mathrm{cm}$
  Height of cylindrical support $\left(\mathrm{h}\right)=7\mathrm{cm}$
  The formula for the curved surface area of the cylinder$=2\mathrm{\pi rh}$

$$\begin{gathered} =2\times \frac{22}{7}\times 1.5\times 7 \\ =66{\mathrm{cm}}^2 \end{gathered}$$
Area to be painted black $=(8\times 66){\mathrm{cm}}^2=528{\mathrm{cm}}^2$

Step 4: Determination of black paint cost

• Given cost for painting $1{\mathrm{cm}}^2$ with black color $\mathrm{Rs}0.05$
  Therefore, the cost of painting in black color $=\mathrm{Rs}(528\times 0.05)$
  $=\mathrm{Rs}26.40$

Step 5: The total painting cost

• Cost of silver paint + Cost of black paint$=\mathrm{Rs}(2757.86+26.40)$

$=\mathrm{Rs}2784.26$

Hence, the total painting cost will be $\mathrm{Rs}.2784.26$.