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

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

Answer:

Cost of paint \(= 2784.26\)

  
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