Question

A circular park, 42 m in diameter, has a path 3.5 m wide running
round it on the outside. Find the cost of gravelling the path at
20 per m
m?​

Answer 1

Answer:

area of path = area of bigger circle - area of smaller circle

radius of bigger circle = 42 + 3.5 = 45.5m

area of bigger circle = πr²

area of bigger circle = 22/7 × 45.5 × 45.5

area of bigger circle = 6506.5 m²

area of smaller circle = πr²

area of smaller circle = 22/7 × 42 × 42

area of smaller circle = 5544 m²

area of path = 6505.5 - 5544

area of path = 962.5 m²

the cost of gravelling the path = area of path × cost of gravelling at per m²

the cost of gravelling the path = 962.5 × 20

the cost of gravelling the path = 19250 rupees

Answer 2

Given :

The diameter of a circular park is 42 m.

A 3.5 m wide path lies along the periphery just outside the park.

$\begin{gathered}\end{gathered}$

To Find :

The cost of constructing a pavement on the path at the rate of $20per m².

$\begin{gathered}\end{gathered}$

Using Formulas :

$\longrightarrow\small{\underline{\boxed{\sf{Radius = \dfrac{Diameter}{2}}}}}$

$\longrightarrow\small{\underline{\boxed{\sf{Area \: of \: Inner \: circle = \pi{r}^{2}}}}}$

$\longrightarrow\small{\underline{\boxed{\sf{Area \: of \: outer\: circle = \pi{R}^{2}}}}}$

${\longrightarrow{\small{\underline{\boxed{\sf{Area_{(parth)} = Area_{(outer\: circle)} - Area_{(inner \: circle)}}}}}}}$

${\longrightarrow{\small{\underline{\boxed{\sf{Cost = Area \: of \: path \times constructing \: rate \: per \: {m}^{2}}}}}}}$

$\begin{gathered}\end{gathered}$

Solution :

⚘ Finding the area of park without path :-

$\longrightarrow{\sf{Radius = \dfrac{Diameter}{2}}}$

$\longrightarrow{\sf{Radius = \dfrac{42}{2}}}$

$\longrightarrow{\sf{Radius = \cancel\dfrac{42}{2}}}$

$\longrightarrow{ \underline{\boxed{\sf{\red{Radius = 21 \: m}}}}}$

∴ The radius of park without path is 21 m.

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⚘ Finding the area of park with path :-

$\longrightarrow{\sf{Radius = 21 + 3.5}}$

$\longrightarrow{\sf{Radius = 24.5 \: m}}$

$\longrightarrow{ \underline{\boxed{\sf{\red{Radius = 24.5\: m}}}}}$

∴ The area of park with path is 24.5 m.

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⚘ Finding area of Inner circle :-

$\longrightarrow{\sf{Area \: of \: Inner \: circle = \pi{r}^{2}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \dfrac{22}{7} \times {(21)}^{2}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \dfrac{22}{7} \times {(21 \times 21)}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \dfrac{22}{7} \times {441}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \dfrac{22 \times 441}{7}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \dfrac{9702}{7}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = \cancel{\dfrac{9702}{7}}}}}$

${\longrightarrow{\sf{Area \: of \: Inner \: circle = 1386 \: {m}^{2}}}}$

${\longrightarrow{ \underline{\boxed{\sf{\red{Area \: of \: Inner \: circle = 1386 \: {m}^{2}}}}}}}$

∴ The area of inner circle is 1386 m².

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⚘ Finding the area of outer circle :-

$\longrightarrow{\sf{Area \: of \: outer\: circle = \pi{R}^{2}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \dfrac{22}{7} \times {(24.5)}^{2}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \dfrac{22}{7} \times {(24.5 \times 24.5)}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \dfrac{22}{7} \times {600.25}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \dfrac{22 \times 600.25}{7}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \dfrac{13205.5}{7}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = \cancel{\dfrac{13205.5}{7}}}}}$

${\longrightarrow{\sf{Area \: of \: outer\: circle = 1886.5 \: {m}^{2}}}}$

${\longrightarrow{ \underline{\boxed{\sf{\red{Area \: of \: outer\: circle = 1886.5\: {m}^{2}}}}}}}$

∴ The area of outer circle is 1886.5 m².

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⚘ Finding the area of path :-

${\longrightarrow{\small{\underline{\boxed{\sf{Area_{(parth)} = Area_{(outer\: circle)} - Area_{(inner \: circle)}}}}}}}$

${\longrightarrow{\sf{Area_{(parth)} = 1886.5 - 1386}}}$

${\longrightarrow{\sf{Area_{(parth)} =500.5 \: {m}^{2}}}}$

${\longrightarrow{\underline{\boxed{\sf{\red{Area \: of \: parth =500.5 \: {m}^{2}}}}}}}$

∴ The area of path is 505.5 m².

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⚘ Now finding the cost of constructing a pavement on the path :-

${\longrightarrow{\sf{Cost = Area \: of \: path \times constructing \: rate \: per \: {m}^{2}}}}$

${\longrightarrow{\sf{Cost = 500.5 \times 20}}}$

${\longrightarrow{\sf{Cost = \ 10010}}}$

${\longrightarrow{\underline{\boxed{\sf{\red{Cost = \ 10010}}}}}}$

∴ The cost of constructing a pavement on the path is $10010.

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