Cups A and B are cone shaped and have heights of #25 cm# and #18 cm# and openings with radii of #8 cm# and #3 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?

The volume of liquid in A = 169.646

To find height of liquid solve:

multiply by 3 to eliminate fraction.

To determine whether 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 given by:

[ V = \frac{1}{3} \pi r^2 h ]

where ( V ) is the volume, ( r ) is the radius of the base, and ( h ) is the height of the cone.

For cup A:

• Height ( h = 25 ) cm
• Radius ( r = 8 ) cm

For cup B:

• Height ( h = 18 ) cm
• Radius ( r = 3 ) cm

Now, we can calculate the volumes of both cups:

For cup A: [ V_A = \frac{1}{3} \pi (8^2) (25) = \frac{1}{3} \pi (64) (25) = \frac{1600}{3} \pi ]

For cup B: [ V_B = \frac{1}{3} \pi (3^2) (18) = \frac{1}{3} \pi (9) (18) = 54 \pi ]

Since ( V_B < V_A ), cup B's contents can fit inside cup A without overflowing. To find out how high cup A will be filled, we need to calculate the height at which the volume of cup A equals the volume of cup B:

[ \frac{1}{3} \pi (8^2) (h) = 54 \pi ] [ \frac{64}{3} \pi h = 54 \pi ] [ h = \frac{54 \pi \cdot 3}{64 \pi} ] [ h \approx \frac{162}{64} \approx 2.53 \text{ cm} ]

So, cup A will be filled to a height of approximately ( 2.53 ) cm when the contents of cup B are poured into it.