Question

The curved surface area of a frustum cone is $25\pi m{m^2}$. The larger circle area is $12\pi m{m^2}$. The total surface area is $350\pi m{m^2}$. Find the smaller circle area of a cone
A) $313m{m^2}$
B) $323\pi m{m^2}$
C) $333\pi m{m^2}$
D) $313\pi m{m^2}$

Answer 1

Hint:
We are given the curved surface area , total surface area and the area of the bigger circle and we can use the formula of area of circle $\pi {r^2}sq.units$, curved surface area of frustum $\pi (r + R)h{\text{ }}sq.units$and obtain the values of ${R^2}{\text{ and }}(R + r)h$ and substitute it in the formula of total surface area of the sphere $\pi [(r + R)h + {r^2} + {R^2}]{\text{ }}sq.units$. And once use the area of the circle formula to get the area of the smaller circle.

Complete step by step solution:
Let's consider a frustum with height h and smaller radius r and bigger radius R

We are given the area of the bigger circle to $12\pi m{m^2}$
We know that the area of circle is given by $\pi {r^2}sq.units$
Therefore
Area of bigger circle $ = \pi {R^2}sq.units$
$
   \Rightarrow 12\pi = \pi {R^2} \\
   \Rightarrow 12 = {R^2} \\
 $
Let this be equation (1)
We are given that the curved surface area of the frustum is $25\pi m{m^2}$
We know that the curved surface area of a frustum is given by $\pi (r + R)h{\text{ }}sq.units$
$
   \Rightarrow \pi (r + R)h{\text{ = 25}}\pi \\
   \Rightarrow (r + R)h = 25 \\
 $
Let this be equation (2)
We are given that the total surface area is $350\pi m{m^2}$
We know that the total surface area of a frustum is given by $\pi [(r + R)h + {r^2} + {R^2}]{\text{ }}sq.units$
$
   \Rightarrow \pi [(r + R)h + {r^2} + {R^2}]{\text{ = 350}}\pi \\
   \Rightarrow [(r + R)h + {r^2} + {R^2}]{\text{ }} = 350 \\
 $
Substitute the values from equation (1) and (2) we get
$
   \Rightarrow (25 + {r^2} + 12) = 350 \\
   \Rightarrow 37 + {r^2} = 350 \\
   \Rightarrow {r^2} = 350 - 37 \\
   \Rightarrow {r^2} = 313 \\
 $
We know that the area of circle is given by $\pi {r^2}sq.units$
Therefore
Area of smaller circle $ = \pi {r^2}sq.units = 313\pi m{m^2}$

The correct option is D.

Note:
A frustum is
1) the part of a conical solid left after cutting off a top portion with a plane parallel to the base.
2) the part of a solid, as a cone or pyramid, between two usually parallel cutting planes.