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=(1)/(2)d=(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=(LateralSurfaceArea)/(\pi*r )`

Substituting, we have:

`l=(211.06m^(2))/(\pi*5 )=(211.06)/(15.71m )`

`l=(211.06)/(15.71m )=13.43m`

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

`l=13.43m`

Have a nice day!