Since it is conical, we need to find the surface area of the top of the conical shape.
If we unfold the top part of the cone, we will have a section of a circle:
The circunference of this section is the same as the total circunference of the base of the cone, which we can get from its radius (half its diamtere):
[tex]C=2\pi r=2\pi\cdot\frac{D}{2}=2\pi\cdot\frac{10}{2}=10\pi[/tex]If we visualize the cone by the side, we see that it forms a isosceles triangle which the same height and the base euqal to the diameter:
So, we can calculate "R", the radius of the unfolded cone, using the Pythagora's Theorem:
[tex]\begin{gathered} R^2=h^2+(\frac{D}{2})^2 \\ R^2=12^2+5^2 \\ R^2=144+25 \\ R^2=169 \\ R=\sqrt[]{169} \\ R=13 \end{gathered}[/tex]The circunference of a section of a circle is the circunferece of the total circle times the fraction of the section represents of the total circle. Let's call ths fraction "f", this means that:
[tex]\begin{gathered} C_{total}=f\cdot C \\ C_{total}=2\pi R=2\pi\cdot13=26\pi \\ C=10\pi \\ f\cdot26\pi=10\pi \\ f=\frac{10\pi}{26\pi}=\frac{5}{13} \end{gathered}[/tex]The area will follow the same, the area of the section is the fraction "f" times the total area of the circle, so:
[tex]\begin{gathered} A_{total}=\pi R^2=\pi13^2=169\pi \\ A=f\cdot A_{total}=\frac{5}{13}\cdot169\pi=65\pi\approx65\cdot3.14=204.1 \end{gathered}[/tex]So, the surface area of the top of the cone is 204.1 ft². Since each skin covers 15 ft², we can calculate how many skins we need by dividing the total by the area of each skin:
[tex]\frac{204.1}{15}=13.60666\ldots[/tex]This means that we need 13.60666... skins, that is, 13 is not enough, we need one more, so we need a total of 14 skins.
