Question

A well , 14 m deep, is 2 m in radius . find the cost of cementing the inner curved surface at the rate of Rs 2 per square metre.​

Answer 1

${\large{\bold{\rm{\underbrace{\underline{Let's \: understand \: the\: concept \: 1^{st}}}}}}}$

☀ This question says that a well is 14 metres deep and it's radius is 2 metres. We have to find the cost of cementing the inner curved surface at the rate of Rs 2 mere per².

☀ As we already know that the well is in shape of cylinder so we have to use the formula to find curved surface area (cylinder) and the cost of cementing the inner curved surface at the rate of Rs 2 mere per² here.

${\large{\bold{\rm{\underline{Given \: that}}}}}$

♛ Deepness of well = 14 m (height)

♛ Radius of well = 2 m

♛ The cost of cementing the inner curved surface at the rate of Rs 2 per m²

${\large{\bold{\rm{\underline{To \: find}}}}}$

♛ The cost of cementing the inner curved surface

${\large{\bold{\rm{\underline{Solution}}}}}$

♛ The cost of cementing the inner curved surface = Rupees 352

${\large{\bold{\rm{\underline{Using \: concept}}}}}$

♛ Formula to find CSA of cylinder

${\large{\bold{\rm{\underline{Using \: formula}}}}}$

${\boxed{\boxed{\sf{CSA_{(cylinder)} \: = \: 2 \pi rh}}}}$

Where,

♛ CSA denotes Curved Surface Area

♛ r denotes radius

♛ h denotes height

♛ π pronounced as pi

♛ Value of π is 3.14 or ${\sf{\dfrac{22}{7}}}$

${\large{\bold{\rm{\underline{Solution}}}}}$

~ Let's find the CSA of well !

${\sf{\longrightarrow CSA_{(cylinder)} \: = \: 2 \pi rh}}$

$\:{\sf{\longrightarrow CSA_{(well)} \: = \: 2 \times \dfrac{22}{7} \times 2 \times 14}}$

$\: \:{\sf{\longrightarrow CSA_{(well)} \: = \: 2 \times \dfrac{22}{7} \times 28}}$

$\: \: \:{\sf{\longrightarrow CSA_{(well)} \: = \: \dfrac{44}{7} \times 28}}$

$\: \: \:{\sf{\longrightarrow CSA_{(well)} \: = \: 44 \times 4}}$

$\: \: \: \:{\sf{\longrightarrow CSA_{(well)} \: = \: 176 m^{2}}}$

${\green{\frak{CSA \: is \: 176 \: m^{2}}}}$

~ Let's find the cost

Let us take a product of 176 and 2

$\: \: \: \:{\sf{\longrightarrow Cost \: = 176 \times 2}}$

$\: \:{\sf{\longrightarrow Cost \: = 352 \: Rupees}}$

${\green{\frak{Cost \: of \: cementing \: is \: 352 \: Rupees}}}$

${\large{\bold{\rm{\underline{Additional \: information}}}}}$

Cylinder diagram -

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\thicklines\multiput(-0.5,-1)(26,0){2}{\line(0,1){40}}\multiput(12.5,-1)(0,3.2){13}{\line(0,1){1.6}}\multiput(12.5,-1)(0,40){2}{\multiput(0,0)(2,0){7}{\line(1,0){1}}}\multiput(0,0)(0,40){2}{\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\multiput(18,2)(0,32){2}{\sf{r}}\put(9,17.5){\sf{h}}\end{picture}$

Some formulas -

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cylinder \: = \: \pi r^{2}h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Surface \: area \: of \: cylinder \: = \: 2 \pi rh + 2 \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Lateral \: area \: of \: cylinder \: = \: 2 \pi rh}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Base \: area \: of \: cylinder \: = \: \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Height \: of \: cylinder \: = \: \dfrac{v}{\pi r^{2}}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Radius \: of \: cylinder \: = \: \dfrac{v}{\pi h}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto TSA \: of \: cuboid \: = \: 2(l \times b + b \times h + l \times h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto LSA \: of \: cuboid \: = \: 2h(l+b)}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cuboid \: = \: L \times B \times H}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Diagonal \: of \: cuboid \: = \: \sqrt 3l}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Perimeter \: of \: cuboid \: = \: 12 \times Sides}}}$

${\sf{Where,}}$

$\; \; \;{\tt{\rightarrow TSA \: denotes \: total \: surface \: area}}$

$\; \; \;{\tt{\rightarrow LSA \: denotes \: lateral surface area}}$

$\; \; \;{\tt{\rightarrow l \: denotes \: length}}$

$\; \; \;{\tt{\rightarrow b \: denotes \: breadth}}$

$\; \; \;{\tt{\rightarrow h \: denotes \: height}}$

$\; \; \;{\tt{\rightarrow \sqrt means \: square \: root}}$