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

## 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

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

**Answer:**

Cost of paint \(= 2784.26\)