Question

A piece of cheese is cut in the shape of a sector of a circle of radius 6cm and the thickness of the cheese is 7cm as shown in the below figure.

Then Find
1. curved surface area of the sphere.
2. Volume of the cheese piece.

Answer 1

Hint: Given that, the piece of cheese is in the form of a sector of a circle. We know that the arc length of the circular sector is given by $l=\dfrac{\theta }{360{}^\circ }\times 2\pi r$ where $r$ is the radius and $\theta $ is the angle made by the sector at the center. Now we will calculate the surface area of the piece by using the area of the rectangle of length $l$ and width/thickness $t$ i.e. $A=l\times t$. Now we will calculate the area of the circular sector by using the formula $a=\dfrac{\theta }{360{}^\circ }\times \pi {{r}^{2}}$ and we will multiply with thickness $t$ to get the value of volume.

Complete step-by-step solution
Given that,
A piece of cheese is cut in the shape of a sector of a circle of radius $6$ cm and the thickness of the cheese is $7$ cm as shown in the figure below.

Here
Radius $r=6$ cm
Angle $\theta =60{}^\circ $
Thickness $t=7$ cm.
Now the arc length of the circular sector is given by
$\begin{align}
  & l=\dfrac{\theta }{360{}^\circ }\times 2\pi r \\
 & \Rightarrow l=\dfrac{60{}^\circ }{360{}^\circ }\times 2\times 3.14\times 6 \\
 & \Rightarrow l=\dfrac{1}{6}\times 6.28\times 6 \\
 & \Rightarrow l=6.28cm \\
\end{align}$
Now the area of the circular sector is given by
$\begin{align}
  & a=\dfrac{\theta }{360{}^\circ }\times \pi {{r}^{2}} \\
 & \Rightarrow a=\dfrac{60{}^\circ }{360{}^\circ }\times 3.14\times 6\times 6 \\
 & \Rightarrow a=18.84c{{m}^{2}} \\
\end{align}$
Now the surface area of the circular sector is equal to area of rectangle having length equal to arc length and width equal to thickness, then
$\begin{align}
  & A=l\times t \\
 & \Rightarrow A=6.28\times 7 \\
 & \Rightarrow A=43.96c{{m}^{2}} \\
\end{align}$
Now the volume of the piece is given by
$\begin{align}
  & V=a\times t \\
 & \Rightarrow V=18.84\times 7 \\
 & \Rightarrow V=131.88c{{m}^{3}} \\
\end{align}$
Now the surface area and volume of the piece are $43.96c{{m}^{2}}$ and $131.88c{{m}^{3}}$ respectively.

Note: We can also directly use the below formulas to get the surface area and volume.
$\begin{align}
  & A=\dfrac{\theta }{360{}^\circ }\times 2\pi rt \\
 & \Rightarrow A=\dfrac{60{}^\circ }{360{}^\circ }\times 2\times 3.14\times 6\times 7 \\
 & \Rightarrow A=43.96c{{m}^{2}} \\
\end{align}$ and $\begin{align}
  & V=\dfrac{\theta }{360{}^\circ }\times \pi {{r}^{2}}t \\
 & \Rightarrow V=\dfrac{60{}^\circ }{360{}^\circ }\times 3.14\times 6\times 6\times 7 \\
 & \Rightarrow V=131.88c{{m}^{3}} \\
\end{align}$
From both the methods we got the same result.