# Cups A and B are cone shaped and have heights of #37 cm# and #27 cm# and openings with radii of #9 cm# and #5 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?

A is bigger in both dimensions, so will hold the contents of B at height

A is bigger than B in both radius and height, so of course B's volume is less and A will not overflow. We have

The height of A after receiving the contents of B is given by

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To determine if cup A will overflow when the contents of cup B are poured into it, compare the volumes of the two cups.

The volume of a cone is given by the formula: (V = \frac{1}{3} \pi r^2 h), where (r) is the radius and (h) is the height.

For cup A: [ V_A = \frac{1}{3} \pi (9)^2 \cdot 37 ]

For cup B: [ V_B = \frac{1}{3} \pi (5)^2 \cdot 27 ]

Calculate both volumes to compare. If (V_B) (the volume of cup B) is less than or equal to (V_A) (the volume of cup A), then cup A will not overflow when the contents of cup B are poured into it. If (V_B) is greater than (V_A), then cup A will overflow.

After determining that cup A will not overflow, to find how high cup A will be filled, subtract the volume of cup B from the volume of cup A, and then divide by the new radius of cup A to find the additional height.

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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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