Ex.13.5 Q8 Surface Areas and Volumes Solution - NCERT Maths Class 9

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Question

A solid cube of side \(12 \rm \,cm\) is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Text Solution

Reasoning:

Volume of the cube is \(\begin{align}\,{a^3} \end{align}\) when the side measurement is a. Surface area is nothing but the sum of the area of the \(6\) faces.

What is  known?

Side length of the cube. Number of cubes cut from the bigger cube.

What is  unknown?

Volume of the small cube and ratio between the surface areas of the bigger and the smaller cube.

Steps:

Side of the solid cube \(= 12\rm cm\)

Volume of the solid cube

\(\begin{align}={a^3} = {(12)^3} = 1728\,\,\rm {m^3} \end{align}\)

It is cut in to \(8\) equal cubes of same volume.

Therefore,

Volume of one small cube

\[\begin{align} & = \frac{{1728}}{8} \\& = 216 \rm \,{m^3} \end{align}\]

Let \(x\) be side of the small cube.

\(\begin{align}\therefore \end{align}\)Volume of one small cube\(\begin{align}\,{x^3}\rm c{m^3} \end{align}\)

\[\begin{align}{x^3} &= 216\\x &= \sqrt[3]{{216}}\\\,\,\,\, &= {(216)^{\frac{1}{3}}}\\\,\,\,\, &= {(6 \times 6 \times 6)^{\frac{1}{3}}}\\\,\,\,\, &= 6\,\,cm \end{align}\]

For the ratio of this surface area we have to find the surface area of the two cubes having side 12cm and 6 cm respectively.

Surface area of the cube \(\begin{align}\, = 6 \times {a^2} \end{align}\)

Surface area of the cube having \(12 \rm cm \) side

\[\begin{align} = 6 \times {12^2} \end{align}\]

Surface area of the cube having \(6\rm cm\) side \(\begin{align}= 6 \times {6^2} \end{align}\)

\[\begin{align} \therefore \rm Ratio& = \frac{{6 \times 12 \times 12}}{{6 \times 6 \times 6}} \\ &= \frac{4}{1} = 4:1 \end{align}\]

Answer:

The side of the new cube is \(6\rm m.\)

Ratio between the surface area \(= 4:1.\)