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??

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Answer 1 · 2018-04-22T17:56:33

See below.

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)#