Question

Please help me find the slant height of the cone and the height of the base. I can try and plug everything in, but please help me! I dont get how to make an equation for this??

Answer 1

See below.

Explanation:

I'm not sure why you are asking for the slant height. This is `bby` in the diagram and it says `bb(y=2)`.

The diagram shows the lateral surface area of a cone, for various values of `x`. The arc length is also the circumference of the base of a cone.

To find the arc length we can use:

`"Arc length"=rtheta`

Where theta is in radians.

Note here we are using a radius of 2, which is the slant height.

We are given degree measurement, but we can covert this to radians in the following way.

`30^@=(30pi)/180=(pi)/6`

For `30^@`:

Circumference is:

`2((pi)/6)=color(blue)(pi/3)`

To find the height of the cone we use Pythagoras' theorem. We first need to find the radius.

Circumference = `2pir`

`:.`

`pi/3=2pir`

`r=(pi)/(6pi)=1/6`

`"Height" = sqrt(y^2-r^2)`

`\ \ \ \ \ \ \ \ \ \ \ \ \=sqrt(4-1/36)=sqrt(143/36)=(sqrt(143))/6=color(blue)(1.99)color(white)(88)` 2 d.p.

The rest can be found in exactly the same manner. I will do the second one and leave you to do the rest.

For `90^@`

Convert:

`90^@=(90pi)/180=pi/2`

Circumference:

`2(pi/2)=color(blue)(pi)`

Radius:

`pi=2pir=>r=1/2`

`"Height"=sqrt(y^2-r^2)=sqrt(4-1/4)=sqrt(15/4)=(sqrt(15))/2=color(blue)(1.94)`