Ex.13.4 Q7 Surface Areas and Volumes Solution - NCERT Maths Class 9

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Question

Q7. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.

Text Solution

Reasoning:

Surface of the sphere having radius \(r\) is equal to the \(4\) times are of the circle having radius \(r.\) \(\begin{align}S = 4\pi {r^2} \end{align}\)

What is known?

Ratio between the diameter of the moon and earth.

What is unknown?

Ratio of the surface areas.

Steps:

Let the diameter of the earth be \(2r.\)

Radius of the earth in \(= r\)

Radius of the moon is \(= \frac{r}{4}\) 

Surface area of the earth \(= 4\pi \rm{r^2}\) 

Surface area of the moon \(= 4 \pi\left(\frac{r}{4}\right)^{2}\)

\[\begin{align}\text{∴ Ratio of their surface area} &=\frac{\text { Surface area of themoon }}{\text { Surface area of the earth }}\\&=\frac{4 \times \pi \times r^{2}}{4 \times \pi \times 4 \times 4 \times r^{2}}\\&=\frac{1}{16}\end{align}\]

Answer:

\(\begin{align}\text{Ratio of their surface area = 1:16}\end{align}\)