Pawan Prajapati
Hint: At first, we need to find the volume of both small and large cones. Then compare with the relation that the volume of the smaller cone is 127
of the larger one. Then use the property of similarity to write r1r=h1h+h1
, then substitute to find the values of h1
and h respectively.
Complete step-by-step answer:
In the question, the height of the cone is given 30 cm. Now it is further said that a small cone is cut off at the top by a plane parallel to the surface of the base. The cone which was cut off is (127)
of the larger original cone. Now, we have to find at which height above the base it should be cut so that the conditions are satisfied. In the given figure,
The height of the bigger cone is 30 cm. So, we can say,
h1+h=30
⇒h1=30−h
We are given in the figure, r is the radius of the larger cone, and r1
is the radius of the smaller cone. So we will find the volume of the smaller and larger cone by using the formula πr2h3
where r is radius and h is the height.
So, the volume of the smaller cone is
13×π×(r1)2×h1
⇒πr21h13
And the volume of the larger cone of height 30 cm is
13×π×(r)2×30=10πr2
We know that the volume of the smaller cone is (127)
of the larger cone.
So, we get,
πr21h13=127(10πr2)
On simplifying further, we get,
r21h1=109r2
It can be further written as,
9h110=r2r21
As the base of the smaller cone whose base was cut parallel to that of the larger cone, then we can say that they are similar to each other. Then we can say,
r1r=h1h+h1
So, we can write,
rr1=h+h1h1
Now, as we know that h+h1=30
. So,
rr1=30h1
As we know that
9h110=r2r21
So, we can substitute rr1=30h1
So, we get,
9h110=900h21
On further cross-multiplication, we get,
9h31=9000
⇒h31=1000
Here, h1=10cm
So, h is equal to 30 – h1 which is (30 – 10) cm = 20 cm.
Hence, the height is 20 cm.
Note: Students should be well versed with the formulas of the volume of the cone. They should also know the properties of similar figures, i.e. if two figures are similar to each other, then their dimensions are in ratio to each other.
