The answer is:
The slant height is 13.43 m.
[tex]l=13.43m[/tex]
To solve the problem, we need to use the following equations to calculate the total surface area and the lateral surface area of right cone:
[tex]TotalSurfaceArea=LateralSurfaceArea+BaseArea[/tex]
[tex]LateralSurfaceArea=\pi *r*l[/tex]
Where,
r, is the radius of the cone.
l, is the slant height of the cone.
We are given the following information:
[tex]TotalSurfaceArea=290.6m^{2} \\Diameter=10m\\Radius=\frac{1}{2}d=\frac{1}{2}10m=5m[/tex]
So, calculating the area of the base(circle) in order to find the lateral surface area, we have:
[tex]BaseArea=\pi *r^{2} \\\\BaseArea=\pi *5m^{2} =\pi *25m^{2}=79.54m^{2}[/tex]
Then, substituting the area of the base into the total surface area to calculate the surface area of the cone, we have:
[tex]LateralSurfaceArea=TotalSurfaceArea-BaseArea[/tex]
[tex]LateralSurfaceArea=290.6m^{2}-79.54m^{2}[/tex]
[tex]LateralSurfaceArea=211.06m^{2}[/tex]
Now, calculating the slant height, we have:
[tex]LateralSurfaceArea=\pi *r*l[/tex]
[tex]l=\frac{LateralSurfaceArea}{\pi*r }[/tex]
Substituting, we have:
[tex]l=\frac{211.06m^{2}}{\pi*5 }=\frac{211.06}{15.71m }[/tex]
[tex]l=\frac{211.06}{15.71m }=13.43m[/tex]
Hence, we have that the slant height is 13.43 m.
[tex]l=13.43m[/tex]
Have a nice day!
