Question

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

Answer 1

Answer:

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

Step-by-step explanation:

Given : An inverted cone of vertical height 12 cm and base radius 9 cm contains water to a depth of 4 cm.

To find: The area of the interior surface of the cone not in contact with the water?

Solution :

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.

So, it forms frustum of cone.

By similarity of triangles as shown in the figure below:

$\frac{OA}{OB}=\frac{AC}{BD}\\\\\frac{4}{12}=\frac{r}{9}\\\\r=\frac{9\times 4}{12}=\frac{36}{12}=3$

And height of frustum = 12-4=8 cm

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':

$l=\sqrt{8^2+(9-3)^2}\\\\l=\sqrt{64+6^2}\\\\l=\sqrt{64+36}\\\\l=\sqrt{100}\\\\l=10$

So, Area is given by

$Area=\frac{22}{7}\times 10\times (9+3)\\\\Area=\frac{22}{7}\times 10\times 12\\\\Area=377.14\ cm^2$

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