Question

The dimensions of a cuboud are in a ratio 3:2:2 and the lateral surface area of the cuboid is 200m². The outer surface of the cuboud is painted with enamel at the rate of ₹10 per m². Find the total cost of painting the outer surface of the cuboid.

Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ total \ cost \ of \ painting \ outer}$

$\sf{surface \ area \ of \ cuboid \ is \ Rs \ 3200}$

$\sf\orange{Given:}$

$\sf{\implies{The \ dimensions \ of \ a \ cuboid \ are}}$

$\sf{in \ ratio \ of \ 3:2:2}$

$\sf{\implies{Latera \ surface \ area \ of \ cuboid=200 \ cm^{2}}}$

$\sf{\implies{The \ outer \ surface \ is \ painted \ at \ rate \ of}}$

$\sf{Rs \ 10 \ per \ m^{2}}$

$\sf\pink{To \ find:}$

$\sf{Cost \ of \ painting \ outer \ surface \ of \ the}$

$\sf{cuboid.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{The \ dimensions \ of \ a \ cuboid \ are \ in}$

$\sf{a \ ratio \ 3:2:2}$

$\sf{Let \ common \ multiple \ be \ x.}$

$\sf{\therefore{Length(l)=3x}}$

$\sf{\therefore{Breadth(b)=2x}}$

$\sf{\therefore{Height(h)=2x}}$

$\sf{Lateral \ surface \ area \ of \ cuboid=2(l+b)h}$

$\sf{...formula}$

$\sf{\therefore{200=2(3x+2x)2x}}$

$\sf{\therefore{5x\times \ 2x=\frac{200}{2}}}$

$\sf{10x^{2}=100}$

$\sf{x^{2}=10}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{x=\sqrt10}$

$\sf{\therefore{Length(l)=3\sqrt10}}$

$\sf{\therefore{Breadth(b)=2\sqrt10}}$

$\sf{\therefore{Height(h)=2\sqrt10}}$

$\sf{Outer \ surface \ area=Total \ surface \ area}$

$\sf{Total \ surface \ area \ of \ cuboid=2(lb+bh+hl)}$

$\sf{...formula}$

$\sf{\implies{2(3\sqrt10\times2\sqrt10+2\sqrt10\times2\sqrt10+3\sqrt10\times2\sqrt10)}}$

$\sf{\implies{2(60+40+60)}}$

$\sf{\implies{2\times160}}$

$\sf{\implies{320 \ cm^{2}}}$

$\sf{Rate \ of \ painting \ is \ Rs \ 10 \ per \ m^{2}}$

$\sf{Cost \ of \ painting \ 320 \ m^{2}=320\times10}$

$\sf{\therefore{Cost \ of \ painting=Rs \ 3200}}$

$\sf\purple{\tt{\therefore{The \ total \ cost \ of \ painting \ outer}}}$

$\sf\purple{\tt{surface \ area \ of \ cuboid \ is \ Rs \ 3200}}$