Question

$\huge\underline{ \sf \color{red}{Question:-}}$
The diameter of a circular park is 75 m. A 3.5 m wide road runs on
the inside around it. Find the cost of constructing the road at ₹200
per square metre.



$\: \: \bull{Any~mod/star/serious~user ~ plz ~answer}$​

Answer 1

Answer :-

$\:\\\large{\underbrace{\sf{Firstly,\;let's\;understand\;the\;concept\;used\;:-}}}$

Here the concept of Areas of Circle has been used. We see first we can find out the area of the circular park. Then we can add the width of the road to the radius of the park and find the whole area including path and park. From this whole area we can subtract the area of path. The we can multiply this area with the rate.

Let's do it !!

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★ Formula Used :-

$\:\\\large{\boxed{\sf{Areas\;of\;Circle\;=\;\bf{\pi r^{2}}}}}$

$\:\\\large{\boxed{\sf{Radius\;of\;whole\;plot_{(including\;park\;and\;path)}\;=\;\bf{Radius\;of\;Park\;+\;Width\;of\;Road}}}}$

$\:\\\large{\boxed{\sf{Area\;of\;Path\;=\;\bf{Area\;of\;whole\;plot\;-\;Area\;of\;Park}}}}$

$\:\\\large{\boxed{\sf{Total\;cost\;of\;constructing\;the\;road\;=\;\bf{Rate_{(in\:per\:m^{2})}\:\times\:Area\;of\;Path}}}}$

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★ Question :-

The diameter of a circular park is 75 m. A 3.5 m wide road runs on the inside around it. Find the cost of constructing the road at ₹200 per square metre.

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★ Solution :-

Given,

» Diameter of circular park = 75 m

» Radius of circular park = ½ × 75 = 37.5 m

» Width of the road = 3.5 m

» Rate of construction of road = Rs. 200 per m²

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~ For the Area of Park :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Areas\;of\;Circle_{(Park)}\;=\;\bf{\pi r^{2}}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Areas\;of\;Circle_{(Park)}\;=\;\bf{\dfrac{22}{7}\:\times\: (37.5)^{2}\:\;=\:\;\underline{\underline{4419.64\;\;m^{2}}}}}}$

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~ For the Radius of whole plot :-

$\:\\\qquad\large{\sf{\rightarrow\;\;\:Radius\;of\;whole\;plot_{(including\;park\;and\;path)}\;=\;\bf{Radius\;of\;Park\;+\;Width\;of\;Road}}}$

$\:\\\qquad\large{\sf{\rightarrow\;\;\:Radius\;of\;whole\;plot_{(including\;park\;and\;path)}\;=\;\bf{37.5\;+\;3.5\:\;=\;\:\underline{\underline{41\;\;m}}}}}$

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~ For the Area of whole plot :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Areas\;of\;Circle_{(whole\;plot)}\;=\;\bf{\pi r^{2}}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Areas\;of\;Circle_{(whole\;plot)}\;=\;\bf{\dfrac{22}{7}\:\times\: (41)^{2}\:\;=\:\;\underline{\underline{5283.14\;\;m^{2}}}}}}$

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~ For the Area of Path :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Area\;of\;Path\;=\;\bf{Area\;of\;whole\;plot\;-\;Area\;of\;Park}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Area\;of\;Path\;=\;\bf{5283.14\;-\;4419.64\;\:=\;\:\underline{\underline{863.5\;\;m^{2}}}}}}$

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~ For the Total Cost of Constructing the Road :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Total\;cost\;of\;constructing\;the\;road\;=\;\bf{Rate_{(in\:per\:m^{2})}\:\times\:Area\;of\;Path}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Total\;cost\;of\;constructing\;the\;road\;=\;\bf{200\;\times\;863.5\:\;=\;\:\underline{\underline{Rs.\;\;172700}}}}}$

$\:\\\large{\underline{\underline{\rm{Thus,\;total\;cost\;of\;construction\;of\;road\;is\;\;\boxed{\bf{Rs.\;\;172700}}}}}}$

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★ Evaluation from the Attachment :-

From attachment we see that, the brown shaded region us the road which is to be constructed around the park and the green shaded portion is the park. Now if we subtract the area of green shaded portion from the area of whole bigger circular region, we can get the area of path.

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★ More to know :-

• Area of Square = (Side)²

• Area of Rectangle = Length × Breadth

• Area of Triangle = ½ × Base × Height

• Area of Parallelogram = Base × Height

• Perimeter of Square = 4 × Side

• Perimeter of Rectangle = 2 × (Length + Breadth)

• Perimeter of Circle = 2πr

Answer 2

Given:-

• $\sf \: Diameter = 75 \: m$

• $\sf \: Radius \: = 37.5 \: m$

• $\sf \: Outer \: radius = 37.5 + 3.5 = 41 \: m$

Solution:-

${ \underline{ \boxed{ \sf{Outer \: area \: from \: center =\pi {r}^{2} }}}}</p><p>$

$\bullet= \sf \: 3.14 \times {41}^{2}$

$\bullet \sf \: = 5278.34 \: {m}^{2}$

${ \underline{ \boxed{ \sf{Inner \: area \: = \: \pi {r}^{2} }}}}$

$\bullet = \sf \: 3.14 \times{37.5}^{2}$

$\bullet\sf \: = \: 4,415.625 \: m {}^{2}$

${ \underline{ \boxed{ \sf{Area \: of \: round = Outer \: area \: - \: Inner \:area}}}}$

$\bullet \sf \: = (5278.34 ) - (4,415.625)$

$\bullet \sf \: = 862.715 \: {m}^{2}$

${ \underline{ \boxed{ \sf{Cost \: of \: construction \: = Rs.2.00 \: {m}^{2} }}}}$

${ \underline{ \boxed{ \sf{cost \: of \: construction \: = Rs.2.00 \times 862.715}}}}$

$\bullet \sf \: = Rs. \: 1,725.43$

The cost of constructing road is Rs.

1,725.43..