Question

The following figure shows a closed victory stand whose dimension are given in cm.
If the bottom of the stand is open, find its volume and total surface area.

• A. $1,72,000c{m}^2$ and $15,000c{m}^2$
• B. $1,42,000c{m}^2$ and $15,000c{m}^2$
• C. $1,62,000c{m}^2$ and $15,000c{m}^2$
• D. $1,32,000c{m}^2$ and $15,000c{m}^2$

Answer 1

The correct option is D $1,32,000c{m}^2$ and $15,000c{m}^2$

Volume of the stand $=$ Volume of left part of stand with dimensions $30cm,40cm,20cm+$ Volume of middle part of stand with dimensions $30cm,40cm,60cm+$ Volume of right part of stand with dimensions $30cm,40cm,30cm$
Since each part is cuboidal, their volume $=length\times breadth\times height$
So, Volume of left part of stand with dimensions $30cm,40cm,20cm=30\times 40\times 20=24,000{cm}^3$
Volume of middle part of stand with dimensions $30cm,40cm,60cm=30\times 40\times 60=72,000{cm}^3$
Volume of right part of stand with dimensions $30cm,40cm,30cm=30\times 40\times 30=36000{cm}^3$
Hence, volume of the victory stand $=24,000+72,000+36000=1,32,000{cm}^3$
Surface area $=$ Sum of surface areas of each face which is visible as the bottom of the stand is open.

We know that surface area of a rectangular face $=dimension1\times dimension2$
So, for the smallest step in the victory stand towards the left:
Surface area of the face facing towards left with dimensions $40cm,20cm=40\times 20=800c{m}^2$
Surface area of the face facing towards front with dimensions $30cm,20cm=30\times 20=600c{m}^2$
The face facing backward will have the same surface area of the face facing front $=600c{m}^2$
Surface area of the face facing towards front with dimensions $30cm,20cm=30\times 20=600c{m}^2$
Surface area of the face facing towards top with dimensions $40cm,30cm=40\times 30=1,200c{m}^2$
For the tallest step in the victory stand, which is in the middle.
The face of the facing towards left will have breadth $=40cm$ and height $=(30+30)-20=40cm$ as the total height of the stand is $30+30=60cm$ of which $20cm$ is not a part of the face.
So, Surface area of the face facing towards left with dimensions $40cm,40cm=40\times 40=1,600c{m}^2$
The face of the facing towards front will have length $=30cm$ and height $=(30+30)=60cm$ as the total height of the stand is $30+30=60cm$
So, Surface area of the face of the facing towards front with dimensions $30cm,60cm=30\times 60=1,800c{m}^2$
The face facing backward will have the same surface area of the face facing front $=1,800c{m}^2$
Surface area of the face of the facing towards top with dimensions $40cm,30cm=40\times 30=1,200c{m}^2$
Surface area of the face facing towards right with dimensions $40cm,30cm=40\times 30=1,200c{m}^2$
For the medium height step in the victory stand towards the right:
Surface area of the face facing towards right with dimensions $40cm,30cm=40\times 30=1,200c{m}^2$
Surface area of the face facing towards front with dimensions $30cm,30cm=30\times 30=900c{m}^2$
The face facing backward will have the same surface area of the face facing front $=900c{m}^2$
Surface area of the face facing towards top with dimensions $40cm,30cm=40\times 30=1,200c{m}^2$
Hence Total Surface area

$=800+600+600+600+1,200+1,600+1,800+1,800+1,200+1,200+1,200+900+900+1,200=15000{cm}^2$