Cups A and B are cone shaped and have heights of #32 cm# and #24 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?
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To determine if cup A will overflow when the contents of cup B are poured into it, we need to compare the volumes of the two cups. Since both cups are cone-shaped, we can use the formula for the volume of a cone, which is ( V = \frac{1}{3}\pi r^2 h ), where ( r ) is the radius of the base and ( h ) is the height.
Let's calculate the volumes of cup A and cup B:
For cup A:
- Radius (( r )) = 9 cm
- Height (( h )) = 32 cm
Volume of cup A: [ V_A = \frac{1}{3}\pi \times (9)^2 \times 32 ]
For cup B:
- Radius (( r )) = 5 cm
- Height (( h )) = 24 cm
Volume of cup B: [ V_B = \frac{1}{3}\pi \times (5)^2 \times 24 ]
Once we have the volumes of both cups, we can compare them. If the volume of cup B is less than or equal to the volume of cup A, then cup A will not overflow. Otherwise, it will overflow.
After comparing the volumes, if cup A is not filled to its maximum capacity, we can calculate how high cup A will be filled by subtracting the volume of cup B from the total capacity of cup A, and then dividing by the area of the base of cup A.
Let's perform the calculations to determine the answers.
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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.
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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