Question

If a cone with a diameter of 10 meters has a surface area of 290.6 square meters, find its slant height.

Answer 1

Hello!

The answer is:

The slant height is 13.43 m.

$l=13.43m$

Why?

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:

$TotalSurfaceArea=LateralSurfaceArea+BaseArea$

$LateralSurfaceArea=\pi *r*l$

Where,

r, is the radius of the cone.

l, is the slant height of the cone.

We are given the following information:

$TotalSurfaceArea=290.6m^{2} \\Diameter=10m\\Radius=\frac{1}{2}d=\frac{1}{2}10m=5m$

So, calculating the area of the base(circle) in order to find the lateral surface area, we have:

$BaseArea=\pi *r^{2} \\\\BaseArea=\pi *5m^{2} =\pi *25m^{2}=79.54m^{2}$

Then, substituting the area of the base into the total surface area to calculate the surface area of the cone, we have:

$LateralSurfaceArea=TotalSurfaceArea-BaseArea$

$LateralSurfaceArea=290.6m^{2}-79.54m^{2}$

$LateralSurfaceArea=211.06m^{2}$

Now, calculating the slant height, we have:

$LateralSurfaceArea=\pi *r*l$

$l=\frac{LateralSurfaceArea}{\pi*r }$

Substituting, we have:

$l=\frac{211.06m^{2}}{\pi*5 }=\frac{211.06}{15.71m }$

$l=\frac{211.06}{15.71m }=13.43m$

Hence, we have that the slant height is 13.43 m.

$l=13.43m$

Have a nice day!