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 , 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 $\pi (r+R)hsq.units$ 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 $\pi [(r+R)h+r^2+R^2]sq.units$. After that use the area of the circle formula to get the area of the smaller circle

Complete step-by-step answer:
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 $=\pi R^{2}sq.units$
$$\begin{gathered} \Rightarrow 12\pi =\pi R^2 \\ \Rightarrow 12=R^2 \end{gathered}$$
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 $\pi (r+R)hsq.units$
$$\begin{gathered} \Rightarrow \pi (r+R)h\text{ = 25}\pi \\ \Rightarrow (r+R)h=25 \end{gathered}$$
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 $\pi [(r+R)h+r^2+R^2]sq.units$
$$\begin{gathered} \Rightarrow \pi [(r+R)h+r^2+R^2]\text{ = 350}\pi \\ \Rightarrow [(r+R)h+r^2+R^2]=350 \end{gathered}$$
Substitute the values from equation (1) and (2) we get
$$\begin{gathered} \Rightarrow (25+r^2+12)=350 \\ \Rightarrow 37+r^2=350 \\ \Rightarrow r^2=350-37 \\ \Rightarrow r^2=313 \end{gathered}$$
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 the part of a conical solid left after cutting off a top portion with a plane parallel to the base. The part of a solid, as a cone or pyramid, between two usually parallel cutting planes.
Many students tend to make the calculation difficult by substituting the value of $\pi$ and then dividing which consumes a lot of time.