the height of a cone is 30 cm. A small cone is cut off at the top by a plane parrallel to the base. If it's volume is 1/27 of the volume of the given cone, at what height above the base is the section made?
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Let the height of the original cone be H and radius be R.
Volume (V1)=(1/3)π(R^2)H
Now, let the height of the small cone be h and radius be r.
It's volume (V2)=(1/3)π(r^2)h
Using similarly of triangles, it can be proved that
r/R = h/H ---- (1)
According to the question;
V2/V1 = 1/27
=>(r^2×h)/(R^2×H)=1/27
=>{(r/R)^2}×(h/H)=1/27
Using equation (1), we get;
{(h/H)^2}×(h/H)=1/27
=>(h/H)^3 = 1/27 = (1/3)^3
=> h/H = 1/3
=> h = H/3
It is given that H=30cm.
Hence h = 10cm.
Therefore, the original cone was cut (30-10)cm above the base.
Required Answer = 20cm.
[Answer]
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Volume (V1)=(1/3)π(R^2)H
Now, let the height of the small cone be h and radius be r.
It's volume (V2)=(1/3)π(r^2)h
Using similarly of triangles, it can be proved that
r/R = h/H ---- (1)
According to the question;
V2/V1 = 1/27
=>(r^2×h)/(R^2×H)=1/27
=>{(r/R)^2}×(h/H)=1/27
Using equation (1), we get;
{(h/H)^2}×(h/H)=1/27
=>(h/H)^3 = 1/27 = (1/3)^3
=> h/H = 1/3
=> h = H/3
It is given that H=30cm.
Hence h = 10cm.
Therefore, the original cone was cut (30-10)cm above the base.
Required Answer = 20cm.
[Answer]
Hope this helps! For clarification or discussion please comment and for more follow me.
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