Question

A path of \[4m\] width runs round a semi-circular glassy plot whose circumference a \[163\dfrac{3}{7}m\].
Find
1.Area of the path
2.The cost of gravelling the path at the rate of \[Rs.1.50\] per square meter.
3.The cost of turfing the plot at the rate of \[45\] paise per \[{{m}^{2}}\].

Answer 1

Hint: Firstly, we will be converting the circumference into improper fraction for our easy calculation. And then we will be finding the radius from the given circumference value by applying its formula. After finding the radius we will be finding the area of the path, cost of gravelling and then the cost of turfing.

Complete step by step answer:
Now let us learn about the circumference and area of a circle. The circumference is nothing but the length of the circle. The circumference of a circle can be calculated by \[2\pi r\] when radius is given. the area of the circle means the area occupied by the circle. This can be calculated by the formula \[\pi {{r}^{2}}\] when the radius is given or known.
Now let us solve the given problem.
Circumference of the circle=\[163\dfrac{3}{7}m\]=\[\dfrac{1144}{7}\]\[m\]
Firstly, we have to find the radius of the circular path from the given circumference.
\[\begin{align}
  & 2\pi r=\dfrac{1144}{7} \\
 & 2\times \dfrac{22}{7}\times r=\dfrac{1144}{7} \\
\end{align}\]
On further solving this, we get
\[\begin{align}
  & 2\times \dfrac{22}{7}\times r=\dfrac{1144}{7} \\
 & r=\dfrac{1144}{7}\times \dfrac{7}{44} \\
 & r=26m \\
\end{align}\]
\[\therefore \] The radius of the circular path is \[r=26m\].
A.Now let us find the area of the path-
Since it is a circular path, we will be applying the area of the ring.
The area of the ring can be calculated by \[\pi \left( {{R}^{2}}-{{r}^{2}} \right)\]
So upon applying this formula, we get
\[\begin{align}
  & \pi \left( {{R}^{2}}-{{r}^{2}} \right) \\
 & \Rightarrow \dfrac{22}{7}\left( {{\left( 26+4 \right)}^{2}}-{{\left( 26 \right)}^{2}} \right) \\
 & =\dfrac{22}{7}\left( 14.97 \right) \\
 & =47.05{{m}^{2}} \\
\end{align}\]
B.Now let us find the cost of gravelling the path at the rate of \[Rs.1.50\] per square meter.
Area to be calculated is = \[47.05\] \[\times 1.50\]= \[Rs.70.575\]
C.Now we will be finding the cost of turfing the plot at the rate of \[45\] paise per \[{{m}^{2}}\].
Area of the plot=\[\pi {{r}^{2}}\]\[=\dfrac{22}{7}\times 26\times 26=2124.5\]
Cost of turfing\[=2124.5\times 0.45=956.025\]

Note: We must have a note that when the cost of plot is being asked, we must consider the entire area of land present and then calculate the cost. And the path would be the area of the ring. We must also have a check upon the units of dimensions. We have used \[\pi =\dfrac{22}{7}\] instead of \[\pi =3.14\] because \[\pi =\dfrac{22}{7}\] is more easier for calculation and give us the accurate answer than \[\pi =3.14\]. Also the values to be considered will depend upon the values of the dimensions given.