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?

Answer 1

The volume of a right circular cone using slant height #l# is...

#V=(1/3)pir^2sqrt(l^2-r^2)#

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

#V=(1/3)pir^2sqrt(12^2-r^2)#

A graph of the volume function is shown below.

Next, take the derivative with respect to r:

#(dV)/(dr)= -(pi r (r^2-96))/sqrt(144-r^2)#

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:

#r=sqrt96=4sqrt6#

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 ...

#h=sqrt(144-r^2)=sqrt(144-96)=sqrt48=4sqrt3#

hope that helps

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7