# Among all right circular cones with a slant height of 12, what are the dimensions (radius and height) that maximize the volume of the cone?

The volume of a right circular cone using slant height

Plugging in the slant height of 12 into the volume equation:

A graph of the volume function is shown below.

Next, take the derivative with respect to r:

Now, find the maximum by setting the above derivative equal to zero and solving for r:

Since r must be a positive number , the only valid solution is:

Verify this is a maximum by checking the first derivative near this solution. To the left, V' is positive and to the right it is negative indicating this is indeed a Maximum.

Finally, the height is ...

hope that helps

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The dimensions that maximize the volume of the cone with a slant height of 12 are: radius = 6 and height = 8.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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