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An inverted cone of height 12 cm and base radius cm contains 20 cm3 of water. Calculate the depth of water in the cone, measured from the vertex.

Answer
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Hint: We will find the volume of the cone with height of 12 cm and base radius cmusing the formula i.e., Volume of a cone=13×π×(radius)2×(height). Similarly, we will consider another cone which is formed due to water. We will calculate its volume by assuming its radius as r and height as h. As the volume of similar figures is in the same ratio as the ratio of the cube of their heights. So, using this and putting the given volume of cone formed by water i.e., 20 cm3 we will find h.

Complete step by step answer:
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As we know, Volume of a cone=13×π×(radius)2×(height)
Consider the larger cone with height 12 cm and base radius cm.
Volume of the larger cone=13×π×(6)2×(12)
=144π
Consider the smaller cone, let its radius be r and height as h.
Given, the cone contains 20 cm3 of water. Therefore,
Volume of the smaller cone=20 cm3
As we know, the volume of similar figures is in the same ratio as the ratio of the cube of their heights. So, we can write
Volume of the larger coneVolume of the smaller cone =(Height of the larger cone)3(Height of the smaller cone)3
On putting the values, we get
144π20 =(12)3(h)3
On cross multiplication, we get
(h)3=20×(12)3144π
On simplification we get
(h)3=20×(12)π
h3=240π
h3=76.39437
h=4.243 cm
Therefore, the depth of water in the cone, measured from the vertex is 4.243 cm.

Note:
We have used the concept of similar figures, but one should keep in mind that two solids are similar if and only if they are the same type of solids and their corresponding linear measures such as radii, height, base, length, etc. are proportional. Students may get confused with the height of the cone and height of the empty part of the cone.