Question

this slant height of a conical tomb is 10.5 if its diameter is 16.8 meters find the cost of construction it at rupees 20 per cubic metre and also the cost of whitewashing its curved surface at rupees 1.5 per square metre

Answer 1

$\small \underline \bold{For \: a \: Conical \: tomb -}$

$\large \underline \bold{Given :-}$

$\: \: \:$$\small \bold{Slant \: height \: (L) = 10.5 \: m}$

$\: \: \:$$\small \bold{diameter \: (d) = 16.8 \: m}$

$\large \underline \bold{To Find :-}$

$\small \bold{1) \: how \: much \: cost \: of \: construction \: it \: at \: 20 \: Rs/m^{3} \: ?}$

$\small \bold{2) \: how \: much \: cost \: of \: whitewashing \: its \: Curved \: surface \: area \: at \: 1.5 \: Rs/m^{2} \: ?}$

$\large \underline \bold{Usable \: Formulas :-}$

$\small \underline \bold{For \: a \: Cone -}$

$\small \bold{1) \: Volume =\dfrac{1}{3}\pi r^{2} h}$

$\small \bold{2) \: TSA = \pi r L}$

$\large \underline \bold{Solution :-}$

$\small \bold{radius \: (r) =\dfrac{d}{2} =\dfrac{16.8}{2} = 8.4 \: m}$

$\small \bold{h =\sqrt{L^{2} - r^{2}}}$

$\small \bold{h =\sqrt{(10.5)^{2} - (8.4)^{2}}}$

$\small \bold{h =\sqrt{(\dfrac{21}{2})^{2} - (\dfrac{42}{5})^{2}}}$

$\small \bold{h =\sqrt{\dfrac{441}{4} - \dfrac{1764}{25}}}$

$\small \bold{h = \sqrt{\dfrac{(441\times 25 - 1764\times 4)}{100}}}$

$\small \bold{h =\sqrt{\dfrac{11025 - 7056}{100}}}$

$\small \bold{h =\sqrt{\dfrac{3969}{100}}}$

$\small \bold{h =\dfrac{63}{10} = 6.3 \: m}$

Firstly ,

$\small \bold{V =\dfrac{1}{3}\times \dfrac{22}{7}\times (\dfrac{42}{5})^{2}\times 6.3}$

$\small \bold{V =\dfrac{22}{\cancel{21}}\times \dfrac{\cancel{42}\times 42}{25}\times 6.3}$

$\small \bold{V = 22\times (\dfrac{2\times 42}{25})\times 6.3}$

$\small \bold{V = 88\times (\dfrac{84}{100})\times 6.3}$

$\small \bold{V = 465.70 \: m^{3}}$

$\small \underline \bold{Cost \: of \: construction -}$

$\: \:$$\small \bold{= 465.70\times 20}$

$\: \:$$\small \bold{= 9314 \: Rs}$

Now ,

$\small \bold{TSA =\dfrac{22}{7}\times 8.4\times 10.5}$

$\small \bold{TSA = 22\times 8.4\times 1.5}$

$\small \bold{TSA = 8.4\times 33}$

$\small \bold{TSA = 277.2 \: m^{2}}$

$\small \underline \bold{Cost \: of \: whitewashing -}$

$\small \bold{= 277.2\times 1.5}$

$\small \bold{= 415.8 \: Rs}$

Answer 2

$\bigstar \underline{\underline{\sf \bf Given:-}}$

• Slant height of a conical tomb = 10.5.
• Diameter = 16.8m.

$\bigstar \underline{\underline{\sf \bf To\ find:-}}$

• The cost of constructing it at Rs.20/m³.
• Cost of white washing its Curved surface at Rs.1.5/m²

✦Formula's used✦

$\mapsto \sf \purple{ Volume\ of \ cone = \frac{1}{3} \pi r^2 h}$

$\\\mapsto \sf \pink{CSA\ of \ cone = \pi rl}$

$\\\mapsto \sf \blue{ l^{2} =r^2+h^2 }$

$\bigstar \underline{\underline{\sf \bf Solution:-}}$

$\setlength{\unitlength}{1.2mm}\begin{picture}(5,5)\thicklines\put(0,0){\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\put(-0.5,-1){\line(1,2){13}}\put(25.5,-1){\line(-1,2){13}}\multiput(12.5,-1)(2,0){7}{\line(1,0){1}}\multiput(12.5,-1)(0,4){7}{\line(0,1){2}}\put(18,1.6){\sf{r}}\put(9.5,10){\sf{h}}\end{picture}$

Given , Diameter = 16.8.

❥ Radius = Diameter / 2

=> 16.8 /2

=> Radius (r) = 8.4m.

Finding Height :-

=> l²=r²+h²

⇒ h² = l²-r²

⇒ h² = (10.5)²-(8.4)²

⇒ h²= 39.69

⇒ h = √39.69

⇒ h = 6.3m.

Volume of cone :-

$:\implies \sf \bold{ Volume\ of \ cone = \frac{1}{3} \pi r^{2} h}\\\\\\:\implies \frac{1}{3} \times \frac{22}{7} \times (8.4)^2 \times 6.3 \\\\ \\:\implies \frac{22}{21} (70.56)(6.3)\\\\\\:\implies \frac{22}{21} (444.528)= \bold{465.69 m^3 }$

Cost of construction :-

Cost of constructing 1m³ = rs.20

Cost of constructing 465.6m³= 465.6×20  = Rs.9313.92

Finding the Curved surface Area :-

$:\implies \sf CSA \ of\ cone = \pi rl\\\\\\:\implies \sf \frac{22}{7} (8.4)(10.5)\\\\\\:\implies \sf 1940.4 / 7\\\\\\:\implies CSA = \bold{277.2m^2.}$

Cost of white washing :-  

Cost of white washing 1m³ = Rs.1.5

Cost of White washing  277.2m² = 1.5 × 277.2= Rs.415.8

---------------------------

✦ AnswEr:-

❥ Cost of constructing it at Rs.20/m³,

$\mapsto \orange{\boxed{ \sf Rs.9313.92 }}$

❥ Cost of whitewashing its Curved surface at Rs.1.5 /m²,

$\mapsto \orange{\boxed{ \sf Rs.415.8 }}$

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