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Ex.13.4 Q9 Surface Areas and Volumes Solution - NCERT Maths Class 9

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Question

A right circular cylinder just encloses a sphere of radius \(r\) (see Fig. 13.22).

Find:

(i) Surface area of the sphere

(ii) Curved surface area of the cylinder

(iii) Ratio of the areas obtained in (i) and (ii).

 Video Solution
Surface-Areas-And-Volumes
Ex exercise-13-4 | Question 9

Text Solution

Reasoning:

The Curved surface area of the cylinder is given by \(\begin{align} = 2\pi rh \end{align}\) and Surface area of the sphere\(\begin{align}\, = 4\pi {r^2} \end{align}\)

What is known?

Radius of the sphere which touches the cylinder.

(i) Surface area of the sphere

Steps:

Radius of the sphere in \(= r\)

Surface area \( = 4\pi \rm{r^2} \)

(ii) Curved surface area of the cylinder.

Steps:

For cylinder: radius\(=r\)

Height \(=2r\)

Curved surface area of the cylinder

\[\begin{align}&= 2\pi rh \\&= 2 \times \pi \times r \times 2r \\&= 4\pi {r^2} \end{align}\]

(iii) Ratio of the areas obtained in (i) and (ii).

\[\begin{align}  &{\text{Ratio of the area}} \\   &\qquad = \frac{{{\text{Surface area of sphere}}}}{{\left( \begin{array}{l}  {\text{Curved surface}} \\   {\text{area of cylinder}} \\   \end{array} \right)}} \\   &\qquad = \frac{{4\pi r^2 }}{{4\pi r^2 }} \\    &\qquad = \frac{1}{1} \\    &\qquad = 1:1 \\   \end{align}\]

Answer:

(i) Surface area of the sphere \(\begin{align} = 4\pi {r^2} \end{align}\)

(ii) Curved surface area of cylinder \(\begin{align} = 4\pi {r^2} \end{align}\)

(iii) Ratio between (i) and (ii) \(\begin{align} = 1:1 \end{align}\)