Let us start this problem by analyzing the area we want to calculate
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To calculate the surface area we will divide the are in two parts:
[tex]\text{ The total area = Area of the base + area of the inner and outer cone}[/tex]
The area of the base:
The area of the base can be calculated as the difference between the areas of the to disks , as follows:
[tex]\begin{gathered} \text{ Area of the base=Area of the outer disk - area of the inner disk} \\ =\pi(12)^2-\pi4^2 \\ \\ =128\pi \\ \end{gathered}[/tex]
Where we use twice the formula for the area of a circle pi*radius^2, for the outer disk the radius is 12 and for the inner disk the radius is 4.
The lateral area of the two cones, the outer and the inner
Now we will calculate the lateral area of a cone (that is we will not include the base) this area is illustrated by the following draw:
The lateral area of a cone can be calculated using the next formula
[tex]\text{ Lateral area of a cone=}\pi r\sqrt{h^2+r^2}[/tex]
Where h is the height of the cone, and r is the radius of the base, for the bigger cone we know from the figure that the height is 6 ft and the radius is 12 ft, for the smaller cone we also know from the figuere that the height is 3 ft and the radius is 4 ft. Therefore we can calculate:
[tex]\begin{gathered} \text{ Lateral area of the bigger cone= }\pi12\sqrt{6^2+12^2} \\ \\ =12\pi\sqrt{180} \end{gathered}[/tex]
and
[tex]\begin{gathered} \text{ Lateral area of the smaller cone= }\pi4\sqrt{3^2+4^2} \\ \\ =4\pi\sqrt{25} \\ \\ =20\pi \end{gathered}[/tex]
Finally, putting all the areas together we find that:
[tex]\begin{gathered} \text{ The total area= The area of teh base+ the lateral area of the two cones} \\ \\ =128\pi+12\sqrt{180}\pi+20\pi \\ \\ =148\pi+12\sqrt{180}\pi \\ \\ =148\pi+72\sqrt{5}\pi \end{gathered}[/tex]
