Cups A and B are cone shaped and have heights of #32 cm# and #12 cm# and openings with radii of #5 cm# and #8 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

Answer 1

When cup B is full and its contents are poured into cup A, cup A will not overflow. It will be filled up to a height of #30.72# cm

Volume of a cone whose radius is #r# and height is #h# is given by #1/3pir^2h#.
Hence volume of Cup A is #1/3pi5^2*32# = #(25*32)/3pi# = #800/3pi# and volume of Cup B is #1/3pi8^(2)*12# = #(64*12)/3pi# = #768/3pi#.

As volume of Cup B is less than that of A, when cup B is full and its contents are poured into cup A, cup A will not overflow.

Let Cup A be filled till height #a#, than #1/3pi5^2*a# = #768/3pi#
or #25a=768# or #a=768/25= 30.72# cm
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Answer 2

To determine if cup A will overflow when cup B's contents are poured into it, we need to compare the volumes of the two cups. If the volume of cup B's contents is greater than or equal to the remaining volume in cup A, then cup A will not overflow. Otherwise, cup A will overflow. We calculate the volumes of the two cups using the formula for the volume of a cone: ( V = \frac{1}{3}\pi r^2 h ), where ( r ) is the radius and ( h ) is the height of the cone.

If cup B's contents are poured into cup A and cup A doesn't overflow, the height to which cup A will be filled is equal to the sum of the original height of cup A and the portion of cup B's contents that fits into cup A without overflowing.

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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.

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