Question

Find the area of the circular path if radius of outer circle is 10m & inner circle is 7m. What will be the cost of cementing it at the cost of 300 Rs per sq. meter?​

Answer 1

Answer :-

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the concept of Areas of Circle has been used. We are already given the figure. First we will find the area or inner circle. Then we will find the area of outer circle. Area of Outer Circle will include area of path added to area of inner circle. Also we know that the path is surrounding the inner circle. So area of path will be the area of outer circle subtracted by area of inner circle.

Let's do it !!

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★ Equations Used :-

$\\\;\boxed{\sf{Area\;of\;Circle\;=\;\bf{\pi r^{2}}}}$

$\\\;\boxed{\sf{Area\;of\;Path\;=\;\bf{Area\;of\;Outer\;Circle\;-\;Area\;of\;Inner\;Circle}}}$

$\\\;\boxed{\sf{Total\;cost\;of\;Cementing\;=\;\bf{Rate_{(in\;per\;m^{2})}\;\times\;Area\;of\;Path}}}$

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★ Solution :-

Given,

» Radius of inner circle = 7 m

» Radius of outer circle = 10 m

» Rate of cementing per m² = Rs. 300

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~ For Area of Inner Circle :-

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{\pi r^{2}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{\dfrac{22}{7}\;\times\;(7)^{2}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{\dfrac{22}{7}\;\times\;7\;\times\;7}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{22\;\times\;7}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{154\;\;m^{2}}}$

$\\\;\underline{\boxed{\tt{Area\;\;of\;\;Inner\;\;Circle\;\;=\;\bf{154\;\;m^{2}}}}}$

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~ For the Area of Outer Circle :-

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Outer)}\;=\;\bf{\pi r^{2}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Outer)}\;=\;\bf{\dfrac{22}{7}\;\times\;(10)^{2}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{\dfrac{22}{7}\;\times\;10\;\times\;10}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{\dfrac{2200}{7}}}$

$\\\;\;\;\;\;\sf{:\rightarrow\;\;\;Area\;of\;Circle_{(Inner)}\;=\;\bf{314.29\;\;m^{2}}}$

This is approximate value came out of rounding off the decimal digits.

$\\\;\underline{\boxed{\tt{Area\;\;of\;\;Outer\;\;Circle\;\;=\;\bf{314.29\;\;m^{2}}}}}$

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~ For the Area of the Path :-

$\\\;\;\;\;\;\sf{:\mapsto\;\;\;Area\;of\;Path\;=\;\bf{Area\;of\;Outer\;Circle\;-\;Area\;of\;Inner\;Circle}}$

$\\\;\;\;\;\;\sf{:\mapsto\;\;\;Area\;of\;Path\;=\;\bf{314.29\;-\;154}}$

$\\\;\;\;\;\;\sf{:\mapsto\;\;\;Area\;of\;Path\;=\;\bf{160.29\;\;m^{2}}}$

$\\\;\large{\underline{\underline{\rm{Thus,\;area\;of\;the\;path\;is\;\;\boxed{\bf{160.29\;\;m^{2}}}}}}}$

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~ For the Total Cost of Cementing the Path :-

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

$\\\;\;\;\;\;\sf{:\Longrightarrow\;\;\;Total\;cost\;of\;Cementing\;=\;\bf{300\;\times\;160.29}}$

$\\\;\;\;\;\;\sf{:\Longrightarrow\;\;\;Total\;cost\;of\;Cementing\;=\;\bf{Rs.\;\;48,087}}$

$\\\;\large{\underline{\underline{\rm{Thus,\;total\;cost\;of\;cementing\;the\;path\;is\;\;\boxed{\bf{Rs.\;\;48,087}}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;\;Area\;of\;Square\;=\;(Side)^{2}}$

$\\\;\sf{\leadsto\;\;\;Area\;of\;Rectangle\;=\;Length\;\times\;Breadth}$

$\\\;\sf{\leadsto\;\;\;Area\;of\;Parallelogram\;=\;Base\;\times\;Height}$

$\\\;\sf{\leadsto\;\;\;Area\;of\;Triangle\;=\;\dfrac{1}{2}\;\times\;Base\;\times\;Height}$

$\\\;\sf{\leadsto\;\;\;Area\;of\;Trapezium\;=\;\dfrac{1}{2}\:\times\:(Sum\;of\;Parallel\;Sides)\:\times\:(Distance\;between\;them)}$

Answer 2

$\huge\bold{\underline{\underline{Answer!!!}}}$

Given,

• Radius of outer circle = 10 m
• Radius of inner circle = 7 m

To find,

• Area of the circular path
• Cost of cementing it at the cost of 300 Rs per sq.meter?

Formula used,

• Area of circle = πr²

Solution,

To find out the area of circular path, firstly let's find the area of both circles.

• Area of inner circle = πr²

${\implies}$ = 22/7 × 7²

${\implies}$ = 154 m²

• Area of outer circle = πr²

${\implies}$ = 22/7 ×10²

${\implies}$ = 314.28 m²

Area of circular path = Area of outer circle - Area of inner circle

${\implies}$ = 314.28 - 154

${\implies}$ = 160.28 m²

Therefore, the area of of circular path is 160.28 m².

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Secondly, let's find out the, Cost of cementing it at the cost of 300 Rs per sq.meter?

• 1 m² = ₹300
• 160.28 m² = ?

Let's cross multiply the numbers.

• 160.28 × 300 = ₹48,084

Therefore, Cost of cementing it at the cost of 300 Rs per sq.meter is ₹48,084.