An inverted cone which is sealed has a liquid up to a height of 4 cm from
the vertex as shown in the figure and the radius of the surface of the liquid
is 3 cm. The radius (in cm) and height (in cm) of the bigger cone are both
integers less than 10. What is the height of the liquid level, if the cone is
inverted to have the base at the bottom?
Answers
Answer:
12 cm is the answer 12 cm is the answer
Given : An inverted cone which is sealed has a liquid up to a height of 4 cm from the vertex and the radius of the surface of the liquid is 3 cm. The radius (in cm) and height (in cm) of the bigger cone are both
integers less than 10.
To Find : height of the liquid level, if the cone is inverted to have the base at the bottom
Solution:
height from vertex = 4 cm
radius of surface of water = 3 cm
Height of cone = h
radius of cone = r
h / 4 = r/ 3
=> h / r = 4 /3
as h and r are integers
hence possible values are
8/6 = 12/9 = .. .. .
but h , r < 10
hence h = 8 and r = 6 are the values
Volume of cone = (1/3)πr²h = (1/3)π6²(8) = 96π cm³
Volume of water = (1/3)π3²(4) = 12π cm³
Hence Space left above water = 96π - 12 π= 84π cm³
A cone of height k is formed above water
h / r = 4 /3 => radius at water surface = 3k/4
whose volume = (1/3)π (3k/4)²k = 84π
=> k³ = 28 * 16
=> k = 4∛7 = 7.65
Height of water level = 8 - 7.65 = 0.35 cm
height of the liquid level, if the cone is inverted to have the base at the bottom = 0.35 cm
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