Answer: Area of the interior surface of the cone not in contact with the water is 377.14 cm² .
Step-by-step 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:
[tex]\frac{OA}{OB}=\frac{AC}{BD}\\\\\frac{4}{12}=\frac{r}{9}\\\\r=\frac{9\times 4}{12}=\frac{36}{12}=3[/tex]
And height of frustum = 12-4=8 cm
As we know the formula for "Curved surface area of frustum ":
[tex]Area=\pi l(r_1+r_2)\\\\where,\\\\l=\sqrt{h^2+(r_1-r_2)^2[/tex]
So, First we find slant height 'l':
[tex]l=\sqrt{8^2+(9-3)^2}\\\\l=\sqrt{64+6^2}\\\\l=\sqrt{64+36}\\\\l=\sqrt{100}\\\\l=10[/tex]
So, Area is given by
[tex]Area=\frac{22}{7}\times 10\times (9+3)\\\\Area=\frac{22}{7}\times 10\times 12\\\\Area=377.14\ cm^2[/tex]
Hence, Area of the interior surface of the cone not in contact with the water is 377.14 cm² .
