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Question 996457: A small cone has dimensions of 4m radius, and 10m height.
The larger cone's dimensions are unknown, but it is 125 times larger than small cone's volume.
How can I find the radius and the height of the larger cone? using the information given.
I tried multiplying the radius and height by 125 but it was wrong :/
please help!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the small cone has 4m radius and 10m height.
volume of a cone formula is 1/3 * pi * r^2 * h
the volume of the small cone is 1/3 * pi * 4^2 * 10.
now the volume of the large cone is equal to 125 times the volume of the small cone.
this can be shown as v(large) = 125 * 1/3 * pi * 4 * 4 * 10.
if you want to keep the proportions correct, you need to apply the same factor to each of the factors of radius and height.
since there are 3 factors (4*4*10), then you need to take the cube root of 125 and apply that to each of them.
you will get:
v(large) = 1/3 * pi * (125^(1/3) * 4) * (125^(1/3) * 4) * (125^(1/3) * 10).
since 125^(1/3) = 5, you get:
v(large) = 1/3 * pi * (5 * 4) * (5 * 4) * (5 * 10), which becomes:
v(large) = 1/3 * pi * 20 * 20 * 50.
the new radius is 20 and the new height is 50.
the large cone dimensions are in proportion with the small cone dimensions.
the volume of the new cone is 1/3 * pi * 20^2 * 50 = 20000/3
the volume of the small cone is 160/3.
the volume of the large cone is 20000/3.
the volume of the new cone is 125 times the volume of the old cone because 20000 / 125 = 160.
basically, you multiplied each component of the small cone by the cube root of 125.
4 * 125^(1/3) = 4 * 5 = 20
10 * 125^(1/3) = 10 * 5 = 50
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