Ex.13.1 Q5 Surface Areas and Volumes - NCERT Maths Class 9

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Question

A cubical box has each edge \(10\,\rm{ cm}\) and another cuboidal box is \(12.5\,\rm{ cm}\) long, \(10\,\rm{ cm}\) wide and \(8\,\rm{ cm}\) high.

(i) Which box has the greater lateral surface area and by how much? 

(ii) Which box has the smaller total surface area and by how much?

 

Text Solution

 

Reasoning:

A cube is cuboid whose length, breadth and height and equal. A cuboid has six faces and the total surface area is the sum of the surface area of the \(6\) faces.

What is the known?

(i) The length of the cube.

(ii) The length, breadth height of the cuboid.

What is the unknown?

Greater lateral surface area and by how much?

Steps:

Lateral surface area of the cube and cuboid is the sum of the area of the four faces.

For cube:

Edge of the cube is \(10\,\rm{ cm}.\)

Lateral surface area of the cube 

\[\begin{align}&=4\,\rm{(edge)^2}\\&= 4 \times {10^2} \\&= 400\,\rm{cm^2} \end{align}\]

For cuboid:

\[\begin{align}\text{length(l)} &= 12.5\,\rm{cm}\\\text{breadth(b)} &= 10\,\rm{cm} \\\text{height(h)} &= 8\,\rm{cm} \end{align}\]

Lateral surface area of the cuboid

\(\begin{align} = 2\,(l + b)\,\,h \end{align}\)

Lateral surface area of the cuboid 

\[\begin{align}&= 2\,\,(12.5 + 10) \times 8\\& = 2(12.5 + 10) \times 8\\& = 2 \times 22.5 \times 8\\ &= 16 \times 22.5\\ &= 360\,\rm{cm^2} \end{align}\]

Cubical box has the greater lateral surface area than the cuboidal box by

\(\begin{align}\,40\,\rm{cm^2}. \;(400 - 360 = 40\,\rm{cm^2}). \end{align}\)

(ii) Smaller total surface area

What is the known?

(i) The length of the cube.

(ii) The length, breadth height of the cuboid.

What is the unknown?

Smaller total surface area and by how much?

Steps:

Total surface area of the cube 

\[\begin{align}&= 6\,\rm{(edge)^2}\\&= 6 \times {10^2}\\&= 600\,\rm{cm^2} \end{align}\]

Total surface area of the cuboid

\[\begin{align}= 2(lb + bh + hl) \end{align}\]

\(\begin{align}&\text{length(l) = 12.5}\,\rm{cm}\\&\text{breadth(b) = 10}\,\rm{cm} \\&\text{height(h) = 8}\,\rm{cm} \end{align}\)

Total surface area: 

\[\begin{align}&=\!2[(12.5\!\times\!10\!+\!10\!\times\!8\!+\!8\!\times\!12.5)]\\&= 2\,\,[125 + 80 + 100]\\ &= 610\,\rm{cm^2} \end{align}\]

Cubical box has the smaller total surface area than the cuboidal box by

\(\begin{align}(610 - 600) = 10\,\rm{cm^2} \end{align}\)

  
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