Question

A conical container can hold 120π cubic centimeters of water. The diameter of the base of the container is 12 centimeters.

The height of the container is ___ centimeters. If its diameter and height were both doubled, the container's capacity would be _____times its original capacity.

Answer 1

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Answer 2

Answer: The height of the container is 10 centimeters. If its diameter and height were both doubled, the container's capacity would be 8 times its original capacity.

Explanation:

The volume of a cone can be calculated with this formula:

`V=(\pi r^2h)/(3)`

Where "r" is the radius and "h" is the height.

We know that the radius is half the diameter. Then:

`r=(12cm)/(2)=6cm`

We know the volume and the radius of the conical container, then we can find "h":

`120\pi cm^3=(\pi (6cm)^2h)/(3)\\\\(3)(120\pi cm^3)=\pi (6cm)^2h\\\\h=(3(120\pi cm^3))/(\pi (6cm)^2)\\\\h=10cm`

The diameter and height doubled are:

`d=12cm*2=24cm\\h=10cm*2=20cm`

Now the radius is:

`r=(24cm)/(2)=12cm`

And the container capacity is

`V=(\pi (12cm)^2(20cm))/(3)=960\pi cm^3`

Then, to compare the capacities, we can divide this new capacity by the original:

`(960\pi cm^3)/(120\pi cm^3)=8`

Therefore, the container's capacity would be 8 times its original capacity.