# Ex.13.9 Q2 Surface Areas and Volumes Solution - NCERT Maths Class 9

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## Question

Q2. The front compound wall of a house is decorated by wooden spheres of diameter $$21\rm\,cm,$$ placed on small supports as shown in Fig 13.32. Eight such spheres are used for this purpose and are to be painted silver. Each support is a cylinder of radius $$1.5 \rm\,cm$$ and height $$7\rm\, cm$$ and is to be painted black. Find the cost of paint required if silver paint costs $$25$$ paise per \begin{align}\rm\,c{m^2} \end{align} and black paint costs $$5$$ paise per \begin{align}\rm\,c{m^2} \end{align}.

## Text Solution

Reasoning:

Surface area of sphere $$=4 \pi r^{2}$$

Surface area of the cylinder $$=2 \pi r h$$

What is  known?

Diameter of the sphere.

Measurement of the cylinder.

Cost per $$\rm\, cm^2$$ for silver and black paint.

What is  unknown?

Cost of the paint.

Steps:

For wooden sphere:

\begin{align} \text{Diameter} = 2r &= 21 \; \rm{mm}\\ &= r = \frac{{21}}{2}\,\,cm \end{align}

Surface area for wooden sphere $$= 4$$

\begin{align} &= 4 \times \frac{{22}}{7} \times {(\frac{{21}}{2})^2}\\ &= 1386\,\,c{m^2} \end{align}

Since the support is in cylinder of radius $$1.5\rm\,cm.$$

So the area of wooden sphere to be painted \begin{align} = 1386 - \frac{{22}}{7}{(1.5)^2} \end{align}[ Area of the sphere is ]

\begin{align} = 1378.93\,\,\rm\,{m^2} \end{align}

Total number of sphere are $$8.$$

Total volume of the sphere to be Silver painted \begin{align} = 8 \times 1378.93 \end{align}

\begin{align} = 11031.44\,\,\rm\,c{m^2} \end{align}

Cost of painting at the rate of $$25$$ paise per \begin{align}\rm\,{m^2} \end{align}.

\begin{align} &= \frac{{11031.44 \times 25}}{{100}}\\ &= 2757.86 \end{align}

For a cylindrical support.

Radius $$(r) =1.5 \rm\,m$$

Height $$(h) =7\rm\, m$$

Surface of the cylindrical support $$=2 \pi r h$$

\begin{align} = \frac{2}{3} \times \frac{{22}}{7} \times 1.5 \times 7 = 66\,\,\rm\,{m^2} \end{align}

Surface area of $$8$$ cylindrical support

\begin{align} = 8 \times 66 = 528\,\,\rm\,c{m^2} \end{align}

Cost of black painting per $$\rm\,cm^2$$ $$= 5$$ paise

\begin{align}{\text{Total cost of black painting}} &= \frac{{528 \times 5}}{{150}}\\ &= 26.40 \\ \text{Cost of paint} &= 2757.856 + 26.40 \\ &= 2784.26 \end{align}

Cost of paint $$= 2784.26$$