Question

An inverted cone of vertical height 12cm and Base radius 9cm contains water to a depth of 4 cm. Find the area of the interior surface of the cone not in contact with the water.

Answer 1

Answer: Area of the interior surface of the cone not in contact with the water is 377.14 cm² .

Explanation:

Since we have given that

Height of an inverted cone = 12 cm

Radius of an inverted cone = 9 cm

According to question, this inverted cone has contained water to a depth of 4 cm.

And we need to find the area of interior surface of the cone not in contact with the water.

So, it forms frustum of cone.

By similarity of triangles as shown in the figure below:

`(OA)/(OB)=(AC)/(BD)\\\\(4)/(12)=(r)/(9)\\\\r=(9* 4)/(12)=(36)/(12)=3`

And height of frustum = 12-4=8 cm

As we know the formula for "Curved surface area of frustum ":

`Area=\pi l(r_1+r_2)\\\\where,\\\\l=\sqrt{h^2+(r_1-r_2)^2`

So, First we find slant height 'l':

`l=√(8^2+(9-3)^2)\\\\l=√(64+6^2)\\\\l=√(64+36)\\\\l=√(100)\\\\l=10`

So, Area is given by

`Area=(22)/(7)* 10* (9+3)\\\\Area=(22)/(7)* 10* 12\\\\Area=377.14\ cm^2`

Hence, Area of the interior surface of the cone not in contact with the water is 377.14 cm² .