Question

a resident welfare association of a society has decided to renovate a local park near their society the park is circular shape in having an area of 1386 they decided to make a path 5m wide along the boundary inside in it find the cost total renovating the park if the cost of renovating the park is 50 perimeter and the cost of construction of path is rs 30 perimeter square​

Answer 1

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Answer 2

Given:

Area of circular park = $1386m^{2}$

Width of path = $5m$

Cost of renovating the park per meter = $Rs 50$

Cost of construction of park per sq. meter = $Rs 30$

To find:

The total cost of renovating the park.

The total cost of construction of the path.

Solution:

The area of a circular park is given. The area of a circle, $A_{1}$ is given by the formula:

$A_{1}=\pi R^{2}$

where, $R$ is the radius of the park.

$1386=\pi R^{2}$

Take $\pi=\frac{22}{7}$

$1386=\frac{22}{7}R^{2}$

$\frac{1386*7}{22}=R^{2}$

$441=R^{2}$

∴ $R=21m$

The width of the path that is to be constructed in the interior of the park is$5m$ whereas the radius of the park is calculated as $21m$.  

Now, we have two circles, with two radii. The circle that includes the path is the bigger circle with a radius $R=21m$ and the circle that excludes the path has a radius $r$. This radius is determined by subtracting the path width from the radius of the outer circle.

Radius of inner circle = Radius of the outer circle - Path width

$r=21-5=16m$

The area of the inner circle having a radius $r$ is given as:

$A_{2}=\pi r^{2}$

$A_{2}=\pi * 16^{2}$

$A_{2}=\frac{22}{7}*256=804.6m^{2}$

We need to find the total cost of construction of the path which requires the calculation of the area of the path. To determine the area of the path, we subtract the area of the smaller circle from the area of the bigger circle.

Area of the path = Area of bigger circle - Area of smaller circle

$Area=1386-804.6=581.4m^{2}$

Total cost of renovating the park = Area of Park × Rate per meter

Total cost = $1386*50=Rs69300$

Total cost of construction of the path = Area of path × Rate per meter

Total cost = $581.4*30=Rs17442$

The total cost of renovating the park is Rs 69300 and the total cost for the construction of the path inside the park is Rs 17442.