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The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume is 127 of the volume of the given cone, at what height above the base is the section mode.

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume is 127
of the volume of the given cone, at what height above the base is the section mode.

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 9723 Points
20 days ago
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.

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